A. Simple Aspects of the Structure of Quantum Mechanics

In Chapters 2 to 16 we have used the formulation of quantum mechanics in terms of wave functions and differential operators. This is but one of many equivalent representations of quantum mechanics. In this appendix we shall brieﬂy review that representation and develop an alternative representation in which state vectors correspond to the wave functions and matrices to the operators. To keep things simple we shall restrict ourselves to systems with discrete energy spectra exempliﬁed on the one-dimensional harmonic oscillator.

A.1 Wave Mechanics

In Section 6.3 the stationary Schrödinger equation h¯ d2 m − + ω2x 2 ϕ = E ϕ (x) 2m dx 2 2 n n n of the harmonic oscillator has been solved. The eigenvalues En were found to be = + 1 ¯ ω En (n 2 )h together with the corresponding eigenfunctions 1 x x 2 h¯ ϕ (x) = √ H exp − , σ = . n πσ n 1/2 n σ σ 2 0 ω ( 02 n!) 0 2 0 m Quite generally, we can write the stationary Schrödinger equation as an eigenvalue equation Hϕn = Enϕn , where the Hamiltonian H – as in classical mechanics – is the sum

H = T + V of the kinetic energy

S. Brandt and H.D. Dahmen, The Picture Book of Quantum Mechanics, 413 DOI 10.1007/978-1-4614-3951-6, © Springer Science+Business Media New York 2012 414 A. Simple Aspects of the Structure of Quantum Mechanics

pˆ2 T = 2m and the potential energy m V = ω2x 2 . 2 The difference to classical mechanics consists in the momentum being given in one-dimensional quantum mechanics by the differential operator h¯ d pˆ = i dx so that the kinetic energy takes the form h¯ 2 d2 T =− . 2m dx 2

Two eigenfunctions ϕn(x), ϕm(x), m = n belonging to different eigenvalues 1 Em = En are orthogonal, i.e., +∞ ϕ∗ ϕ = m(x) n(x)dx 0. −∞ Conventionally, for m = n the eigenfunctions are normalized to one, i.e., +∞ ϕ∗ ϕ = n (x) n(x)dx 1, −∞ so that we may summarize +∞ ϕ∗ ϕ = δ m(x) n(x)dx mn , −∞ where we have used the Kronecker symbol 1, m = n δ = . mn 0, m = n The inﬁnite set of mutually orthogonal and normalized eigenfunctions ϕn(x), n = 0, 1, 2, ...,formsacomplete orthonormal basis of all complex- valued functions f (x) which are square integrable, i.e., +∞ f ∗(x) f (x)dx = N 2 , N < ∞ . −∞ N is called the norm of the function f (x). Functions with norm N can be normalized to one,

1 The functions ϕn(x) are real functions. We add an asterisk (indicating the complex conjugate) to the function ϕm(x) in the integral, since in other cases one often has to deal with complex functions. A.2 Matrix Mechanics in an Inﬁnite Vector Space 415

+∞ ϕ∗(x)ϕ(x)dx = 1, −∞ by dividing them by the normalization factor N, 1 ϕ(x) = f (x). N

The completeness of the set ϕn(x), n = 0, 1, 2,..., allows the expansion ∞ f (x) = fnϕn(x). n=0 Because of the orthonormality of the eigenfunctions the complex coefficients fn are simply +∞ = ϕ∗ fn n (x) f (x)dx . −∞ We also get +∞ ∞ 2 ∗ 2 N = f (x) f (x)dx = | fn| . −∞ n=0 The superposition of two normalizable functions ∞ ∞ f (x) = fnϕn(x), g(x) = gnϕn(x) n=0 n=0 with complex coefficients α, β may be expressed by ∞ α f (x) + βg(x) = (α fn + βgn)ϕn(x). n=0 Their scalar product is defined as +∞ ∞ ∗ = ∗ g (x) f (x)dx gn fn . −∞ n=0

A.2 Matrix Mechanics in an Inﬁnite Vector Space

The normalizable functions f (x)formalinear vector space of inﬁnite di- mensionality, i.e., each function f (x) can be represented by a vector f in that space, f (x) → f . 416 A. Simple Aspects of the Structure of Quantum Mechanics

With the base vectors ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 1 ⎟ ⎜ 0 ⎟ ϕ = ⎜ ⎟ ϕ = ⎜ ⎟ ϕ = ⎜ ⎟ ... 0 ⎝ 0 ⎠ , 1 ⎝ 0 ⎠ , 2 ⎝ 1 ⎠ , ...... a general vector f takes the form ⎛ ⎞ f0 ∞ ⎜ ⎟ ⎜ f1 ⎟ f = fnϕ = ⎜ ⎟ . n ⎝ f2 ⎠ n=0 . .

The axioms of the inﬁnite space of complex column vectors are the natural extension of the ones for ﬁnite complex vectors:

(i) Linear superposition (α, β complex numbers): ⎛ ⎞ α f + βg ⎜ 0 0 ⎟ ⎜ α f1 + βg1 ⎟ αf + βg = ⎜ ⎟ . ⎝ α f2 + βg2 ⎠ . .

(ii) Scalar product: ⎛ ⎞ f0 ⎜ ⎟ ∞ + ∗ ∗ ∗ ⎜ f1 ⎟ ∗ g · f = (g , g , g ,...)⎜ ⎟ = g fn . 0 1 2 ⎝ f2 ⎠ n . n=0 .

+ + = ∗ ∗ ∗ ... Here the adjoint g of the vector g has been introduced, g (g0 , g1 , g2 , ), ∗ ∗ ∗ ... as the line vector of the complex conjugates g0 , g1 , g2 , of the components of the column vector ⎛ ⎞ g ⎜ 0 ⎟ ⎜ g1 ⎟ g = ⎜ ⎟ . ⎝ g2 ⎠ . . Because of the inﬁnity of the set of natural numbers an additional axiom has to be added: A.2 Matrix Mechanics in an Inﬁnite Vector Space 417

(iii) The norm |f| of the vectors f is ﬁnite, ∞ | |2 = + · = ∗ = 2 < ∞ f f f fn fn N , N , n=0 i.e., the inﬁnite sum has to converge. Because of Schwartz’s inequality

|g+ · f|≤|g||f|

all scalar products of vectors f, g of the space are ﬁnite.

As in the ordinary ﬁnite-dimensional vector spaces we call a linear transformation A of a function f (x) into a function g(x),

g(x) = Af(x), the linear operator A. Examples of linear transformations are • the momentum operator pˆ =−ih¯ d/dx, h¯ d f pfˆ = (x), i dx

• the Hamiltonian H =−(h¯ 2/2m)d2/dx 2 + V (x),

h¯ 2 d2 f (x) Hf =− + V (x) f (x), 2m dx 2

• the position operator xˆ = x,

xfˆ = xf(x).

Linear operators can be represented by matrices. We show this by the following argument. The function g is represented by the coefﬁcients gm, ∞ +∞ = ϕ = ϕ∗ g(x) gm m(x), gm m(x)g(x)dx . −∞ m=0 The image function g of f is given by ∞ ∞ g(x) = Af(x) = A fnϕn(x) = Aϕn(x) fn , n=0 n=0 i.e., by a linear combination of the images Aϕn of the elements ϕn of the orthonormal basis. The Aϕn themselves can be represented by a linear combination of the basis vectors 418 A. Simple Aspects of the Structure of Quantum Mechanics

∞ Aϕn = ϕm(x)Amn , n = 0, 1, 2, ... , m=0 with the coefﬁcients +∞ = ϕ∗ ϕ Amn m(x)A n(x)dx . −∞ Inserting this into the expression for g(x) we obtain ∞ ∞ g(x) = ϕm(x)Amn fn . m=0 n=0

Comparing with the representation for g(x) we ﬁnd for the coefﬁcients gm the expression ∞ gm = Amn fn . n=0

We arrange the coefficients Amn like matrix elements in an infinite matrix scheme ⎛ ⎞ A A A ··· ⎜ 00 01 02 ⎟ ⎜ A10 A11 A12 ··· ⎟ A = ⎜ ⎟ ⎝ A20 A21 A22 ··· ⎠ ...... and recover an infinite dimensional extension of the matrix multiplication g = Af in the form ⎛ ⎞ ⎛ ⎞⎛ ⎞ ⎛ $ ⎞ ··· ∞ g0 A00 A01 A02 f0 $n=0 A0n fn ⎜ ⎟ ⎜ ··· ⎟⎜ ⎟ ⎜ ∞ ⎟ ⎜ g1 ⎟ ⎜ A10 A11 A12 ⎟⎜ f1 ⎟ ⎜ $n=0 A1n fn ⎟ ⎜ ⎟ = ⎜ ··· ⎟⎜ ⎟ = ⎜ ∞ ⎟ . ⎝ g2 ⎠ ⎝ A20 A21 A22 ⎠⎝ f2 ⎠ ⎝ n=0 A2n fn ⎠ ...... It should be noted that the two descriptions by wave functions and operators or by vectors and matrices are equivalent. The correspondence relations ⎛ ⎞ a0 ∞ ⎜ ⎟ ⎜ a1 ⎟ ϕ(x) = anϕn(x) ↔ ϕ = ⎜ ⎟ ⎝ a2 ⎠ n=0 . . with +∞ = ϕ∗ ϕ an n (x) (x)dx −∞ A.3 Matrix Representation of the Harmonic Oscillator 419 for wave functions and vectors work in both directions. For a given wave function ϕ(x) we can uniquely determine the vector ϕ relative to the basis ϕn(x), n = 0, 1, 2, ... . Conversely, for a given vector ϕ relative to the basis ϕn(x) we can reconstruct the wave function ϕ(x) as the above superposition of the ϕn(x). The descriptions in terms of ϕ(x)andϕ contain the same information about the state the system is in. Thus, generally one does not distinguish the two descriptions and says the system is in the state ϕ, often denoted by the ket |ϕ as introduced by Paul A. M. Dirac. The wave function ϕ(x)orthevectorϕ are considered merely as two representations out of which many can be invented. The same statements hold true for the representation of operators in terms of differential operators or matrices. Also these are only representations of one and the same linear transformation called linear operator. The states ϕ like the wave functions or vectors ϕ form a linear vector space with scalar product. This general space is called Hilbert space. The linear operators transform a state of the Hilbert space into another state.

A.3 Matrix Representation of the Harmonic Oscillator

Since the ϕn(x) are normalized eigenfunctions of the Hamiltonian, we ﬁnd for its matrix elements +∞ = ϕ∗ ϕ Hmn m(x)H n(x)dx −∞ +∞ = + 1 ¯ ω ϕ∗ ϕ = + 1 ¯ ωδ (n 2 )h m(x) n(x)dx (n 2 )h mn . −∞ Thus, the matrix representation of the Hamiltonian of the harmonic oscillator in its eigenfunction basis is diagonal: ⎛ ⎞ 1 ... 2 00 ⎜ 3 ... ⎟ ⎜ 0 2 0 ⎟ H = h¯ ω ⎜ 5 ... ⎟ . ⎝ 002 ⎠ ......

The representation of the eigenfunctions ϕn(x) in their own basis are given by the standard columns ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ 1 0 0 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ 0 ⎟ ⎜ 1 ⎟ ⎜ 0 ⎟ ϕ = ⎜ ⎟ ϕ = ⎜ ⎟ ϕ = ⎜ ⎟ ... 0 ⎝ 0 ⎠ , 1 ⎝ 0 ⎠ , 2 ⎝ 1 ⎠ , ...... 420 A. Simple Aspects of the Structure of Quantum Mechanics

Of course, the eigenvector equation is recovered also in matrix representation, ϕ = + 1 ¯ ωϕ H n (n 2 )h n . With the help of the recurrence relations for Hermite polynomials,

dHn(x) = 2nH − (x), dx n 1 Hn+1(x) = 2xHn(x) − 2nHn−1(x), we ﬁnd the matrix representations for the position operator xˆ and the momentum operator pˆ in harmonic-oscillator representation, σ ! 2 0 1 x xˆϕ = xϕ (x) = √ nH − (x) + H + (x) exp − n n n 1/2 n 1 2 n 1 2 ( πσ02 n!) 2σ √ √ 0 σ0 = √ n ϕn−1(x) + n + 1 ϕn+1(x) . 2

The coefficients xmn are given by +∞ σ √ √ = ϕ∗ ϕ = √0 δ + + δ xmn m(x)x n(x)dx n m(n−1) n 1 m(n+1) , −∞ 2 and the matrix representation of the position operator is ⎛ ⎞ 0100√ ... ⎜ ⎟ ⎜ 10 20... ⎟ σ ⎜ √ √ ⎟ = √0 ⎜ 0 20 3 ... ⎟ x ⎜ √ ⎟ . 2 ⎝ 00 30... ⎠ ...... For the momentum operator we obtain in a similar way the matrix representation ⎛ ⎞ 0i√ 0 0... ⎜ ⎟ ⎜ −i0√ i20√ ... ⎟ h¯ ⎜ ⎟ p = √ ⎜ 0 −i 20i√ 3 ... ⎟ . σ ⎜ ⎟ 2 0 ⎝ 00−i 30... ⎠ ...... One easily verifies the commutation relation h¯ [p, x] = px − xp = i also for the infinite matrix representations of x and pˆ. Both matrices for x and pˆ are Hermitean, i.e., ∗ = ∗ = xnm xmn , pnm pmn . A.4 Time-Dependent Schrödinger Equation 421

A.4 Time-Dependent Schrödinger Equation

The time dependence of wave functions is determined by the time-dependent Schrödinger equation ∂ ih¯ ψ(x,t) = Hψ(x,t). ∂t

The eigenstates ϕn(x) of the Hamiltonian are the space-dependent factors in an ansatz i ψ (x,t) = exp − E t ϕ (x), n h¯ n n and the eigenvalues En determine the time dependence of the phase factor. In the matrix representation of the harmonic oscillator the time-dependent Schrödinger equation simply reads d ih¯ ψ(t) = Hψ(t), dt where ψ(t) is a vector in Hilbert space, ⎛ ⎞ ψ (t) ⎜ 0 ⎟ ⎜ ψ1(t) ⎟ ψ(t) = ⎜ ⎟ . ⎝ ψ2(t) ⎠ . .

Because of the linearity of the Schrödinger equation any linear combination ∞ ∞ i ψ(x,t) = a ψ (x,t) = a exp − E t ϕ (x) n n n h¯ n n n=0 n=0 also solves the Schrödinger equation. In vectorial representation we have

∞ i ψ(t) = a exp − E t ϕ , E = (n + 1 )h¯ ω . n h¯ n n n 2 n=0 The initial condition at t = 0 for the time-dependent Schrödinger equation is the initial wave function ∞ ψ(x,0)= ψi(x) = ψinϕn(x). n=0 ψ In vector notation this is an initial state vector i. Its decomposition into ϕ eigenvectors n, 422 A. Simple Aspects of the Structure of Quantum Mechanics ⎛ ⎞ ⎛ ⎞ ψi0 a0 ⎜ ⎟ ∞ ⎜ ⎟ ⎜ ψi1 ⎟ ⎜ a1 ⎟ ⎜ ⎟ = ψ = anϕ = ⎜ ⎟ , ⎝ ψi2 ⎠ i n ⎝ a2 ⎠ . n=0 . . . directly provides the identiﬁcation of the expansion coefﬁcients an with the ψ ψ components in of the initial vector i,

an = ψin . This way the time-dependent Schrödinger equation is solved by the expression ∞ i ψ(t) = ψ exp − E t ϕ in h¯ n n n=0 for the initial condition ψ = ψ (0) i . ψ ψ The time-dependent vector n(t) corresponding to n(x,t)is i i ψ (t) = exp − E t ϕ = exp − Ht ϕ , n h¯ n n h¯ n ϕ where En is the energy eigenvalue corresponding to the eigenvector n, i.e., = + 1 ¯ ω En (n 2 )h for the harmonic oscillator. The last equality is meaningful if we deﬁne the exponential of a matrix by its Taylor series ∞ i 1 i n exp − Ht = − Ht . h¯ n! h¯ n=0 For the case of the diagonal matrix H the nth power is trivial, ⎛ ⎞ E n 00... ⎜ 0 ⎟ E n ... n ⎜ 0 1 0 ⎟ H = ⎜ n ... ⎟ , ⎝ 00E2 ⎠ ...... and the explicit matrix form is ⎛ * + ⎞ exp − i E t 00... ⎜ h¯ 0 * + ⎟ ⎜ ⎟ ⎜ − i ... ⎟ i ⎜ 0exph¯ E1t 0 ⎟ exp − Ht = ⎜ * + ⎟ . h¯ ⎜ 00exp− i E t ... ⎟ ⎝ h¯ 2 ⎠ ...... A.5 Probability Interpretation 423

Using the operator representation of i i ψ (x,t) = exp − E t ϕ = exp − Ht ϕ n h¯ n n h¯ n as derived above we may rewrite ψ(t) into the form

∞ i ψ(t) = ψ exp − E ϕ in h¯ n n n=0 ∞ i = exp − Ht ψ ϕ h¯ in n n=0 i = exp − Ht ψ(0) h¯

= UH (t)ψ(0) .

The operator i U (t) = exp − Ht H h¯ is called the temporal-evolution operator.

A.5 Probability Interpretation

ϕ ϕ The eigenfunctions n(x), equivalently the eigenvectors n, describe a state of the physical system with the energy eigenvalue En. Thus, a precise measurement of the energy of this system in the state ϕn should be devised to produce as a result the value En. In order to preserve the reproducibility of the measurement it should not change the eigenstate ϕn of the system during the measurement, i.e., immediately after the energy measurement the state of the system should still be ϕn. The question arises what result will be found in the same energy measurement carried out at a system in a state ϕ described by the wave function ϕ(x), or equivalently, by the vector ϕ being a superposition of eigenfunctions ϕn(x) ϕ or eigenvectors n, ∞ ϕ = anϕn , n=0 with norm one, i.e., ∞ 2 |an| = 1. n=0 424 A. Simple Aspects of the Structure of Quantum Mechanics

The single measurement of the energy will result in one of the energy eigenvalues which we call Em. Reproducibility of the measurement then requires that the system is in the state ϕm after the measurement. 2 The absolute square |am| of the coefﬁcients am in the superposition of the ϕm deﬁning ϕ is the probability with which the energy eigenvalue Em will be determined in the single measurement. Let us assume that we prepare a large assembly of N identical systems, all in the same state ϕ. If we carry out single measurements on these various identical systems we shall measure the energy eigenvalue Em with the 2 abundance |am| N. Performing a weighted average over the results of all measurements yields the expectation value of the energy

∞ ∞ 1 E = |a |2 NE = |a |2 E . N n n n n n=0 n=0 Using the state-vector representation of ϕ, ⎛ ⎞ a ⎜ 0 ⎟ ⎜ a1 ⎟ ϕ = ⎜ ⎟ , ⎝ a2 ⎠ . . we ﬁnd that the energy expectation value is simply ⎛ ⎞⎛ ⎞ E 00... a ⎜ 0 ⎟⎜ 0 ⎟ ⎜ 0 E1 0 ... ⎟⎜ a1 ⎟ ϕ+ Hϕ = (a∗,a∗,a∗,...)⎜ ⎟⎜ ⎟ 0 1 2 ⎝ 00E2 ... ⎠⎝ a2 ⎠ ...... ∞ = ∗ = an Enan E . n=0 Equivalently, in wave-function formulation, we have +∞ +∞ ∞ ∗ ∗ ϕ (x)Hϕ(x)dx = ϕ (x) Enanϕn(x) −∞ −∞ = n 0 ∞ +∞ ∗ = Enan ϕ (x)ϕn(x)dx −∞ n=0 ∞ = ∗ = Enanan E . n=0 B. Two-Level System

In Appendix A the equivalence of wave-function and matrix representation of quantum mechanics was shown. The simplest matrix structure is the one in two dimensions, i.e., in a space with two base states: 1 0 η = , η− = . 1 0 1 1

The linear space consists of all linear combinations χ χ = χ η + χ η = 1 1 1 −1 −1 χ−1 of the base states with complex coefﬁcients χ1 and χ−1. The two states η1 and η−1 form an orthonormal basis of this space, i.e., η+ · η = η+ · η = η+ · η = η+ · η = 1 1 1, −1 −1 1, 1 −1 −1 1 0.

For the linear combination χ to be normalized to one we have

χ + · χ = χ ∗χ + χ ∗ χ =|χ |2 +|χ |2 = 1 1 −1 −1 1 −1 1.

This suggests a representation of the absolute values |χr |, r = 1,−1, of the complex coefﬁcients by trigonometric functions: Θ Θ |χ |=cos , |χ− |=sin . 1 2 1 2 The use of the half-angle Θ/2 is a convention, the usefulness of which will become obvious in the sequel. The complex coefﬁcients themselves are obtained by multiplication of the moduli |χr | with arbitrary phase factors: Θ Θ −iΦ /2 −iΦ− /2 χ = e 1 cos , χ− = e 1 sin . 1 2 1 2 Since a common phase factor is irrelevant the general form can be restricted to

S. Brandt and H.D. Dahmen, The Picture Book of Quantum Mechanics, 425 DOI 10.1007/978-1-4614-3951-6, © Springer Science+Business Media New York 2012 426 B. Two-Level System

Θ Θ −iΦ/2 iΦ/2 χ = e cos , χ− = e sin 1 2 1 2 with Φ = (Φ1 − Φ−1)/2. The general linear combination is therefore Θ Θ −iΦ/2 Θ −iΦ/2 iΦ/2 e cos 2 χ(Θ,Φ) = e cos η + e sin η− = . 2 1 2 1 iΦ/2 Θ e sin 2 The operators corresponding to physical quantities are Hermitean matrices A A − A = 1,1 1, 1 . A−1,1 A−1,−1 The Hermitean conjugate of A is deﬁned as ∗ ∗ + = A1,1 A−1,1 + = ∗ A ∗ ∗ , i.e., Ars Asr . A1,−1 A−1,−1 The condition of Hermiticity, + = ∗ = A A , i.e., Asr Ars , requires ∗ = ∗ = A1,1 A1,1 , A1,−1 A−1,1 , ∗ = ∗ = A−1,1 A1,−1 , A−1,−1 A−1,−1 .

Thus, the diagonal elements A1,1, A−1,−1 are real quantities, the off-diagonal elements A1,−1, A−1,1 are complex conjugates of each other. Hermiticity of the operator A ensures that the expectation value of A for a given general state is real, χ + χ = χ ∗ χ A i Aij j i, j=1,−1 = χ ∗ χ + χ ∗ χ + χ ∗ ∗ χ + χ ∗ χ 1 A1,1 1 1 A1,−1 −1 −1 A1,−1 1 −1 A−1,−1 −1 .

All Hermitean matrices can be linearly combined as the superpositions

A = a0σ0 + a1σ1 + a2σ2 + a3σ3

(with real coefﬁcients a0, ..., a3) of the unit matrix σ = 10 0 01 and the Pauli matrices B. Two-Level System 427 01 0 −i 10 σ = , σ = , σ = , 1 10 2 i0 3 0 −1 since the four matrices σ0, ..., σ3 are Hermitean. One directly veriﬁes the relations σ 2 = σ = i 0 , i 0, 1, 2, 3 , and σ1σ2 = iσ3 , σ2σ3 = iσ1 , σ3σ1 = iσ2 . These yield the commutation relations

[σ1,σ2] = σ1σ2 − σ2σ1 = 2iσ3 and cyclic permutations. The three Pauli matrices can be grouped into a vector in three dimensions,

σ = (σ1,σ2,σ3), with the square σ 2 = σ 2 + σ 2 + σ 2 = σ 2 1 2 3 3 0 . η η σ The base states 1, −1 are eigenstates of the Pauli matrix 3 and of the sum of their squares σ 2,

σ η = η σ 2η = σ η = η = − 3 r r r , r 3 0 r 3 r , r 1, 1, since σ3 and σ0 are diagonal matrices. According to Section A the time-dependent Schrödinger equation reads d ih¯ ξ(t) = Hξ(t), dt where the Hamiltonian is a Hermitean 2 × 2matrix, H H − H = 1,1 1, 1 , H−1,1 H−1,−1 with real diagonal matrix elements H1,1, H−1,−1 and with off-diagonal ele- = ∗ σ ments H−1,1 H1,−1. It can be represented by a superposition of the matrices, H = h0σ0 + h3σ3 + h1σ1 + h2σ2 , where 1 1 h = (H + H− − ), h = (H − H− − ), 0 2 1,1 1, 1 3 2 1,1 1, 1 and 428 B. Two-Level System

1 h = (H − + H− ) = Re H − , 1 2 1, 1 1,1 1, 1 i h = (H − − H− ) =−Im H − . 2 2 1, 1 1,1 1, 1

Introducing the hi (i = 0, 1, 2, 3) into the matrix H we obtain h + h h − ih H = 0 3 1 2 . h1 + ih2 h0 − h3

Introducing the factorization i ξ (t) = exp − E t χ , r = 1,−1, r h¯ r r χ into the time-dependent phase factor and the stationary state r we obtain the stationary Schrödinger equation

χ = χ = − H r Er r , r 1, 1, χ for the eigenstate r belonging to the energy eigenvalue Er . For the eigenval- uesweﬁnd = ±| | | |= 2 + 2 + 2 E±1 h0 h , h h1 h2 h3 .

Since there are only two eigenvalues our system is called a two-level system. The eigenstates are √ 1 |h|+h e−iΦ/2 χ = √ √ 3 , 1 | | |h|−h eiΦ/2 2 h √ 3 − | |− −iΦ/2 χ = √ 1 √ h h3 e −1 | |+ iΦ/2 , 2|h| h h3 e with the phase factor determined by h + ih ei2Φ = 1 2 . h1 − ih2 Introducing the angle Θ by Θ |h|+h Θ |h|−h cos = 3 ,sin= 3 2 2|h| 2 2|h| we may write the eigenstates in the form B. Two-Level System 429

Θ Θ −iΦ/2 iΦ/2 χ = e cos η + e sin η− , 1 2 1 2 1 Θ Θ −iΦ/2 iΦ/2 χ − =−e sin η + e cos η− . 1 2 1 2 1 They are normalized and orthogonal to each other. χ χ The eigenstates 1, −1 of the two-level system exhibit a time dependence which is given by a phase factor only, i ξ (t) = exp − E t χ , r = 1,−1. r h¯ r r

If initially the system is not in an eigenstate the state oscillates. We assume that the initial state is ϕ = η (0) −1 . Decomposition into the eigenstates yields η = ζ χ + ζ χ −1 1 1 −1 −1 with Θ + −iΦ/2 ζ = χ · η− = e sin , 1 1 1 2 Θ + −iΦ/2 ζ− = χ · η− = e cos . 1 −1 1 2 The time-dependent state is obtained as ϕ = ζ ξ + ζ ξ (t) 1 1(t) −1 −1(t) Θ Θ −iΦ/2 −iω1t −iΦ/2 −iω−1t = e sin e χ + e cos e χ − 2 1 2 1 with the angular frequencies

ωr = Er /h¯ , r = 1,−1. η η The probability to ﬁnd the system (originally in the state −1) in the state 1 is given by | | 2 2 h P − = sin Θ sin t , 1, 1 h¯ η and, of course, the probability to ﬁnd it in the state −1 is

P−1,−1 = 1 − P1,−1 . C. Analyzing Amplitude

C.1 Classical Considerations: Phase-Space Analysis

We consider a detector which is capable of measuring position and momentum of a particle simultaneously with certain accuracies. If the result of a measurement is the pair xD, pD of values we may assume that the true values x, p of the quantities to be measured are described by the uncorrelated bivariate Gaussian (cf. Section 3.5) probability density 1 1 (x − x )2 (p − p )2 ρ (x, p, x , p ) = exp − D + D . D D D πσ σ σ 2 σ 2 2 xD pD 2 xD pD That is to say, the probability for the true values x, p of the particle to be the intervals between x and x + dx and between p and p + dp is

dP = ρD(x, p)dx dp . The particle to be measured by the detector possesses position and momentum values x and p. The particle may have been produced by a source which does not deﬁne exactly the values of x and p but according to a probability density 1 1 (x − x )2 (p − p )2 ρ (x, p, x , p ) = exp − S + S . S S S πσ σ σ 2 σ 2 2 xS pS 2 xS pS This is an uncorrelated bivariate Gaussian probability density with the expec- σ 2 σ 2 tation values xS and pS and the variances xS and pS. We now describe how much information can at best be obtained about the probability density ρS(x, p) using the above detector. The probability for a particle prepared by the source to be detected within the intervals (xD, xD + dxD)and(pD, pD + dpD)isgivenby

cl dP = w (xD, pD, xS, pS)dxD dpD with the probability density in phase space

S. Brandt and H.D. Dahmen, The Picture Book of Quantum Mechanics, 430 DOI 10.1007/978-1-4614-3951-6, © Springer Science+Business Media New York 2012 C.1 Classical Considerations: Phase-Space Analysis 431

cl w (xD, pD, xS, pS) +∞ +∞ = ρcl ρcl D (x, p, xD, pD) S (x, p, xS, pS)dx dp −∞ −∞ 1 1 (x − x )2 (p − p )2 = exp − D S + D S . πσ σ σ 2 σ 2 2 x p 2 x p

σ 2 σ 2 Here the variances x and p are obtained by summing up the variances of the detector and source distribution,

σ 2 = σ 2 + σ 2 σ 2 = σ 2 + σ 2 x xD xS , p pD pS .

cl The quantity w (xD, pD, xS, pS) is the result of analyzing the phase-space probability density of the source ρS(x, p, xS, pS) with the help of the phase- cl space probability density ρD(x, p, xD, pD). The function w (xD, pD, xS, pS)is itself a phase-space probability density and obtained through a process we shall call phase-space analysis. This distribution can be measured in principle if the source consecutively produces a large number of particles which are observed in the detector. For a detector of arbitrarily high precision,

σxD → 0, σpD → 0,

cl the distribution w approaches the source distribution ρS(x = xD, p = pD, xS, pS). We have seen that with a detector of high precision and with a sufﬁciently high number of measurements the source distribution can be measured with arbitrary high accuracy. We now assume that the minimum-uncertainty relations, h¯ h¯ σ σ = , σ σ = , xD pD 2 xS pS 2 hold for the widths characterizing the detector and the source. Besides this restriction we stay within the framework of classical physics. Now it is no longer possible to measure the source distribution exactly.1 However, we may still measure the distribution in position alone or the distribution in momentum alone with arbitrary accuracy. To show this we construct the marginal cl distributions of w in the variables xD − xS,andpD − pS, respectively, 1 1 (x − x )2 wcl(x , x ) = √ exp − D S x D S σ 2 2πσx 2 x and

1 cl We could, however, compute the source distribution ρS by unfolding it from w . 432 C. Analyzing Amplitude

Fig.C.1. Phase-space distributions ρS (top), ρD (middle), and their product ρSρD (bottom) together with the marginal distributions of ρS and ρD. The two columns differ only in the spatial mean xS of ρS. Units are used in which h¯ = 1.

1 1 (p − p )2 wcl(p , p ) = √ exp − D S . p D S σ 2 2πσp 2 p The ﬁrst distribution approaches the corresponding marginal position distribution of the source, +∞ 1 1 (x − x )2 ρ (x, x ) = ρ (x, p, x , p )dp = √ exp − S Sx S S S S σ 2 −∞ 2πσxS 2 xS in the case of σxD → 0. However, because of the minimum uncertainty relations, σpD as well as σp approach inﬁnity. Therefore, the second distribution C.1 Classical Considerations: Phase-Space Analysis 433

Fig.C.2. The phase-space distribution ρS (top), the distribution ρD for a particular point (xD, pD) of mean values (middle), and convolution of ρS with ρD for all possible mean values (bottom). Also shown are the marginal distributions. The two columns differ in the widths of ρS.Unitsh¯ = 1 are used. wcl p becomes so wide – and actually approaches zero – that no information about the momentum distribution can be obtained from it. Conversely, for σpD → 0 the momentum distribution of the source can be measured accurately. However, then the information about the position distribution is lost. We illustrate the concepts of this section in Figures C.1 and C.2. We begin with the discussion of the result of a single measurement, yielding the pair xD, pD of measured values. In the two columns of Figure C.1 we show (from top to bottom) the probability density ρS(x, p) characterizing the particle as pro- 434 C. Analyzing Amplitude

duced by the source, the density ρD(x, p) characterizing the detector for the case xD = 0, pD = 0, and the product function ρS(x, p)ρD(x, p). The integral over the product function is the probability density wcl. It is essentially different from zero only if there is a region, the overlap region, in which both ρS and ρD are different from zero. In the left-hand column of Figure C.1 ρS and cl ρD were chosen to be identical, so that w is large. In the right-hand column the overlap is smaller. By very many repeated measurements, each yielding a different result cl xD, pD we obtain the probability density w (xD, pD). In the two columns of Figure C.2 we show (from top to bottom) the probability density ρS(x, p) characterizing the particle, the density ρD(x, p) characterizing the detector for the particular set measured values xD = 0, pD = 0, and the probability cl density w (xD, pD) for measuring the pair of values xD, pD. Also shown are ρ ρ wcl ρ the marginal distribution Sx (x), Dx (x), and x (xD) in position, and Sp(p), ρ wcl Dp(p), and p (pD) in momentum. Comparing in the left-hand column the cl diagram of ρS with the diagram of w we see that the latter distribution is ap- preciably broader than the former in both variables. In the right-hand column, however, the spatial width of the detector distribution σxD isverysmallatthe expense of the momentum width σpD = h¯ /(2σxD), which is very large. The cl distribution w is practically identical to ρS what concerns its spatial varia- tion. The width in momentum of wcl is, however, very much larger than that of ρS.

C.2 Analyzing Amplitude: Free Particle

Quantum-mechanically we describe a particle by the minimum-uncertainty wave packet − 2 ϕ = ϕ = 1 −(x xS) + i − S(x) S(x, xS, pS) / exp pS(x xS) . π 1/4σ 1 2 σ 2 h¯ (2 ) xS 4 xS We consider this wave packet as having been prepared by some physical ap- paratus, the source. The question now arises how the phase-space analysis of the particle as discussed in the last section can be described in quantum mechanics. If, in a particle detector with position-measurement uncertainty σxD and momentum-measurement uncertainty σpD = h¯ /(2σxD), the values xD, pD are measured, we want to interpret the result as in Section C.1. The same probability density 1 1 (x − x )2 (p − p )2 ρ (x, p, x , p ) = exp − D + D D D D πσ σ σ 2 σ 2 2 xD pD 2 xD pD C.2 Analyzing Amplitude: Free Particle 435 describes the probability density of position x and momentum p of the particle. Quantum-mechanically this probability density is the phase-space distribution introduced by Eugene P. Wigner in 1932 of the wave packet − 2 ϕ = 1 − (x xD) + i − D(x, xD, pD) / exp pD(x xD) . π 1/4σ 1 2 σ 2 h¯ (2 ) xD 4 xD

Therefore, ρD(x, p, xD, pD) is also called Wigner distribution of ϕD (cf. Ap- pendix D). Let us construct the analyzing amplitude

+∞ = √1 ϕ∗ ϕ a(xD, pD, xS, pS) D(x, xD, pD) S(x, xS, pS)dx h −∞ representing the overlap between the particle’s wave function ϕS and the de- tecting wave function ϕD. It turns out to be 2 2 1 (xD − xS) (pD − pS) a(xD, pD, xS, pS) = exp − − 2πσ σ 4σ 2 4σ 2 x p x p i σ 2 p + σ 2 p − xD D xS S (x − x ) , σ 2 D S h¯ x where, like in Section C.1,

σ 2 = σ 2 + σ 2 σ 2 = σ 2 + σ 2 x xD xS , p pD pS . The absolute square of the analyzing amplitude 1 1 (x − x )2 (p − p )2 |a|2 = exp − D S + D S πσ σ σ 2 σ 2 2 x p 2 x p cl = w (xD, pD, xS, pS)

cl is identical to the probability density w (xD, pD, xS, pS) of Section C.1. We conclude that the probability amplitude analyzing the values xD and pD of position and momentum of a particle as a result of the interaction of a particle with a detector is given by

+∞ = √1 ϕ∗ ϕ a(xD, pD, xS, pS) D(x, xD, pD) S(x, xS, pS)dx . h −∞

Here, ϕS is the wave function of the particle and ϕD the analyzing wave function. The probability to observe a position in the interval between pD and pD + dpD is 2 dP =|a(xD, pD, xS, pS)| dxD dpD . 436 C. Analyzing Amplitude

In analogy to the classical case we may now ask whether we can still recover the original quantum-mechanical spatial probability density 1 1 (x − x )2 ρ (x) =|ϕ (x)|2 = √ exp − S S S σ 2 2πσxS 2 xS from |a|2. Information about the position of the particle only is obtained by 2 integrating |a| over all values of pD, i.e., by forming the marginal distribution with respect to xD, +∞ | |2 = | |2 = wcl a x a dpD x (xD, xS). −∞

The result is the same as in the classical case. Again in the limit σxD → 0 | |2 we find that the function a x approaches the quantum-mechanical probability density ρS(x) which is equal to the classical distribution ρSx (x). In Figures C.3 and C.4 we demonstrate the construction of the analyzing amplitude using particular numerical examples. Each column of three plots in the two figures is one example. At the top of the column the particle wave function is shown as two curves depicting ReϕS(x)andImϕS(x) together with the numerical values of the parameters xS, pS, σxS which define ϕ(x). Like- wise, the middle plot shows the detector wave function, given by ReϕD(x) and ImϕD(x). The bottom plot contains the real and imaginary parts of the product function ϕ∗ ϕ D(x) S(x), which after integration and absolute squaring yields the probability density 1 +∞ 2 | |2 = ϕ∗ ϕ a D(x) S(x)dx h −∞ of detection. Also given in the bottom plot is the numerical value of |a|2. Four different situations are shown in the two figures. In each case the same detector function ϕD is used. Only the particle wave function ϕS changes from case to case.

(i) In the left-hand column of Figure C.3 ϕS and ϕD are identical. For that case we know that the overlap integral is explicitly real and that +∞ ϕ∗ϕ = | |2 = / = / π = −∞ S S dx 1, so that a 1 h 1 2 in the units h¯ 1 used. (ii) In the right-hand column of Figure C.3 the particle wave packet is moved to a position expectation value xS = xD, but we still have pS = pD, σxS = σxD. By construction, the overlap function is different fromzerointhatx region where both ϕS and ϕD are sizably different from zero. As expected, the value of |a|2 is considerably smaller than in case (i). C.2 Analyzing Amplitude: Free Particle 437

ϕ ϕ ϕ∗ ϕ Fig.C.3. Wave function S (top) and D (middle) and the product function D S (bottom). Real parts are drawn as thick lines, imaginary parts as thin lines. The two columns differ in the mean value xS of ϕS.Unitsh¯ = 1 are used.

(iii) In the left-hand column of Figure C.4 the position expectation values and the widths of particle and detector wave function are identical, xS = xD, σxS = σxD, but the momentum expectation values differ, pS = pD. ϕ∗ ϕ As in case (i) the product function D S is different from zero in the region x ≈ x0 but due to the different momentum expectation values it oscillates. Therefore, the value of |a|2 is much smaller than in case (i) since positive and negative regions of the product function nearly cancel when the integration is performed. 438 C. Analyzing Amplitude

Fig.C.4. As Figure C.3 but for different functions ϕS. The two columns differ only in the value of σxS.

(iv) In the right-hand column of Figure C.4 the particle wave packet has a larger width σxS >σxD. All other parameters are as in case (iii). The product function is similar to that for case (iii) and is concentrated in the region xD −σxD ≤ x ≤ xD +σxD where both wave functions are apprecia- bly different from zero. However, the amplitude of the product function is smaller than in case (iii) since the amplitude of ϕS(x)issmallerinthe overlap region. Therefore, the value of |a|2 is also smaller. C.4 Analyzing Amplitude: Harmonic Oscillator 439

C.3 Analyzing Amplitude: General Case

The lesson learned in the last section can be generalized to the analysis of an arbitrary normalized wave function ϕ(x) describing a single particle in terms of an arbitrary complete or overcomplete set of normalized wave functions ϕ(x)orϕ(x,q1,...,qN ). The functions ϕn(x) can in particular be eigenfunctions of a Hermitean operator, e.g., the energy. Examples for a set of overcomplete functions ϕ(x,q1,...,qn)are

• free wave packets ϕD(x, xD, pD) as in the last section,

• coherent states of the harmonic oscillator ϕ(x, x0, p0) as we shall study in detail in the next section, and

• minimum-uncertainty states of a set of noncommuting operators like the operators L x , L y, L z of angular momentum or Sx ,Sy,Sz of spin as investigated in Sections 10.5 and 17.2.

The analyzing amplitude for the different cases is given by

+∞ = 1 ϕ∗ ϕ a n (x) (x)dx N1 −∞ or +∞ 1 ∗ a = ϕ (x,q1,...,qN )ϕ(x)dx . N2 −∞ Of course, a mutual analysis of two sets of analyzing functions is also of interest, e.g., +∞ = 1 ϕ∗ ϕ ... a n (x) (x,q1, ,qN )dx . N3 −∞ The normalization constants have to be individually determined for every type of analyzing amplitude.

C.4 Analyzing Amplitude: Harmonic Oscillator

For the harmonic oscillator of frequency ω we have discussed in Sections 6.3 and 6.4 two sets of states in particular:

(i) The eigenstates ϕn corresponding to the energy eigenvalues En = (n + 1 ¯ ω 2 )h , √ 2 n −1/2 x x ϕn(x) = ( 2π2 n!σ0) Hn exp − , σ0 2σ0 440 C. Analyzing Amplitude

with the ground-state width √ h¯ σ = 2σ , σ = . 0 x x 2mω

Plots of the ϕn are shown in Figure 6.5. (ii) The coherent states, ∞ i ψ(x,t, x , p ) = a (x , p )ϕ (x)exp − E t , 0 0 m 0 0 m h¯ m m=0

where the complex coefﬁcients an are given by n z 1 ∗ an(x0, p0) = √ exp − z z , n = 0, 1, 2,... . n! 2 The variable z is complex and a dimensionless linear combination of the initial expectation values x0 of position and p0 of momentum,

x0 p0 h¯ z = + i , σp = . 2σx 2σp 2σx Plots of the coherent states ψ(x,t) are shown in Figure 6.6c.

The set of energy eigenfunctions ϕn(x) is complete, the set of coherent states is overcomplete. We can form four kinds of analyzing amplitudes.

Eigenstate – Eigenstate Analyzing Amplitude

We analyze the energy eigenfunctions using energy eigenfunctions as analyzing wave functions. Thus, we obtain as analyzing amplitude +∞

amn = ϕm(x)ϕn(x)dx = δmn , −∞ which yields as probability 2 = δ amn mn .

This result, based on the orthonormality of the eigenfunctions ϕn(x), is illus- ϕ ϕ ϕ2 trated in Figure C.5 which shows the functions m n.Whereas m is non- ϕ2 negative everywhere so that the integral over m cannot vanish it is qualita- tively clear from the ﬁgure that the integral over ϕmϕn vanishes for m = n. The analysis of an eigenstate ϕn(x) with all eigenstates ϕm(x) thus yields 2 = with probability amn 1 the answer that the original wave function was indeed ϕn, and with probability amn = 0 the result that the original wave function was ϕm with m = n. Such an analysis can also be considered as an energy determination which with certainty yields the energy eigenvalue En. C.4 Analyzing Amplitude: Harmonic Oscillator 441

Fig.C.5. Product ϕm (x)ϕn(x) of the wave functions of the harmonic oscillator for h¯ ω = 1. The long-dash curve indicates the potential energy V (x), the short-dash lines show the energy eigenvalue En of the functions ϕn. These lines also serve as zero lines for the product functions.

Eigenstate – Coherent-State Analyzing Amplitude

The function to be analyzed is the time-dependent wave function ψ(x,t, x0, p0) of the coherent state. The analyzing function is the energy eigenfunction ϕn(x). As analyzing amplitude we obtain +∞

a(n, x0, p0) = ϕn(x)ψ(x,t, x0, p0)dx −∞ n z 1 ∗ i = √ exp − z z exp − Ent . n! 2 h¯ The corresponding probability is given by

2 n 2 2 (|z| ) 2 x p |a(n, x , p )|2 = e−|z| , |z|2 = 0 + 0 . 0 0 σ 2 σ 2 n! 4 x 4 p

2 The probabilities |a(n, x0, p0)| for ﬁxed x0, p0 are distributed in the integer n according to a Poisson distribution, cf. Appendix G. Its physical interpretation can be understood if we express |z|2 in terms of the expectation value of the total energy of an oscillator with initial values x0 and p0, 442 C. Analyzing Amplitude

p2 m E = 0 + ω2x 2 . 0 2m 2 0 We ﬁnd 2 |z| = E0/(h¯ ω) = n0 , 2 i.e., |z| equals the number n0 of energy quanta h¯ ω making up the energy E0 of the classical oscillator. This number, of course, need not be an integer. For the absolute square of the analyzing amplitude we thus ﬁnd

n n − |a (x , p )|2 = 0 e n0 . n 0 0 n! It is the probability of a Poisson distribution for the number of energy quanta n found when analyzing a coherent wave function with the eigenfunctions ϕn. It has the expectation value n =n0 and the variance var(n) = n0 .

Coherent-State – Eigenstate Analyzing Amplitude

Analyzing the eigenstate wave functions ϕn(x) with the coherent state wave functions for t = 0, ∞ ϕD(x, xD, pD) = an(xD, pD)ϕn(x), n=0 with the coefﬁcients zn 1 x p = √D − ∗ = D + D an(xD, pD) exp zDzD , zD i , n! 2 2σx 2σp we ﬁnd as the analyzing amplitude

+∞ = √1 ϕ∗ ϕ a(xD, pD,n) D(x, xD, pD) n(x)dx h −∞ n 1 zD 1 ∗ = √ √ exp − z zD , h n! 2 D and for its absolute square

| |2 a(xD, pD,n) n 1 1 x 2 p2 x 2 p2 = √ √ D + D exp − D + D . σ 2 σ 2 σ 2 σ 2 2π( 2σx )( 2σp) n! 4 x 4 p 4 x 4 p C.4 Analyzing Amplitude: Harmonic Oscillator 443

2 Fig.C.6. Absolute square |a(xD, pD,n)| of the amplitude analyzing the harmonic-oscillator eigenstate ϕn(x) with a coherent state of position and momentum expectation value xD and pD.

For a given quantum number n of the eigenstate, |a|2 is a probability density in the xD, pD phase space of the analyzing coherent state which is shown in Figure C.6 for a few values of n. It has the form of a ring wall with the maximum probability at

x 2 p2 |z |2 = D + D = n . D σ 2 σ 2 4 x 4 p In terms of the energy

p2 m E = D + ω2x 2 D 2m 2 D of a classical particle of mass m with position xD and momentum pD in a harmonic oscillator of angular frequency ω,wehave

x 2 p2 |z |2 = D + D = n , D σ 2 σ 2 D 4 x 4 p where nD is the average number of energy quanta h¯ ω in the analyzing wave function ϕD(x, xD, pD). We ﬁnd 444 C. Analyzing Amplitude

n 1 − n |a(x , p ,n)|2 = e nD D . D D h n!

For a given eigenstate ϕn(x) of the harmonic oscillator the probability density in the xD, pD phase space of coherent states depends only on the average number nD of quanta in the analyzing coherent state. The expectation value of nD is given by +∞ +∞ 2 nD = nD|a(xD, pD,n)| dxD dpD = n + 1. −∞ −∞ Its variance has the same value:

var(nD) = n + 1.

Coherent-State – Coherent-State Analyzing Amplitude

Using as analyzing wave functions the coherent states ϕD(x, xD, pD), the analyzing amplitude for the time-dependent coherent states ψ(x,t, x0, p0) turns out to be

a(xD, pD, x0, p0,t) +∞ = √1 ϕ∗ ψ D(x, xD, pD) (x,t, x0, p0)dx h −∞ ! 1 1 ∗ ∗ ∗ i = √ exp − z zD + 2z z(t) + z (t)z(t) exp − ωt h 2 D D 2 with x p z(t) = ze−iωt , z = 0 + i 0 . 2σx 2σp The absolute square yields

| |2 a(xD, pD, x0, p0,t) 1 1 (x − x (t))2 (p − p (t))2 = √ √ exp − D 0 + D 0 σ 2 σ 2 2π( 2σx )( 2σp) 2 2 x 2 p with p x (t) = x cosωt + 0 sinωt , 0 0 mω p0(t) =−mωx0 sinωt + p0 cosωt representing the expectation values of position and momentum of the coherent state ψ(x,t, x0, p0)attimet. It is a bivariate Gaussian in the space of xD and pD centered about the classical positions x0(t), p0(t) of the oscillator. The C.4 Analyzing Amplitude: Harmonic Oscillator 445

2 probability density |a(xD, pD, x0, p0,t)| shows the same behavior as that of a classical particle. The expectation values of the position xD and of momentum pD are simply given by the classical values

xD =x0(t), pD =p0(t).

The variances of xD and pD are

= σ 2 var(xD) 2 x , = σ 2 var( pD) 2 p .

This is twice the values of the ones of the coherent state itself, a consequence of the broadening caused by the analyzing wave packet ϕD(x) having itself σ 2 σ 2 the variances x and p . The classical Gaussian phase-space probability density corresponding to 2 ψ(x,t, x0, p0) is of the same form as |a(xD, pD, x0, p0,t)| . It possesses, however, the widths σx and σp, and has the explicit form

ρcl (x, p, x0, p0,t) 1 1 (x − x (t))2 (p − p (t))2 = exp − 0 + 0 . πσ σ σ 2 σ 2 2 x p 2 x p

By the same token the classical phase-space density corresponding to the de- tecting wave packet is

ρcl D (x, p, xD, pD) 1 1 (x − x )2 (p − p )2 = exp − D + D . πσ σ σ 2 σ 2 2 x p 2 x p

ρcl ρcl The functions and D are equal to the Wigner distributions (cf. Ap- 2 pendix D) of ϕ and ϕD, respectively. The analyzing probability density |a| can again be written as

2 |a(xD, pD, x0, p0,t)| +∞ +∞ = ρcl ρcl D (x, p, xD, pD) (x, p, x0, p0,t)dxD dpD , −∞ −∞ which once more shows the reason for the broadening of |a|2 relative to ρ. D. Wigner Distribution

The quantum-mechanical analog to a classical phase-space probability density is a distribution introduced by Eugene P. Wigner in 1932. In the simple case of a one-dimensional system described by a wave function ϕ(x)the Wigner distribution is deﬁned by 1 +∞ i y y W(x, p) = exp py ϕ(x − )ϕ∗(x + )dy . h −∞ h¯ 2 2

For an uncorrelated Gaussian wave packet with the wave function 2 1 (x − x0) p0 ϕ(x, x0, p0) = √ √ exp − + i (x − x0) 4 σ 2 2π σx 4 x h¯ it has the form of a bivariate normalized Gaussian: 1 1 (x − x )2 (p − p )2 W(x, p, x , p ) = exp − 0 + 0 , 0 0 πσ σ σ 2 σ 2 2 x p 2 x p where σx and σp fulﬁll the minimum-uncertainty relation

σx σp = h¯ /2.

The expression obtained for W(x, p, x0, p0) coincides with the classical phase- space probability density for a single particle introduced in Section 3.6. The marginal distributions in x or p of the Wigner distribution are

+∞ ∗ Wp(x) = W(x, p)dp = ϕ (x)ϕ(x) = ρ(x) −∞ and +∞ ∗ Wx (p) = W(x, p)dx = )ϕ (p))ϕ(p) = ρp(p), −∞ where )ϕ(p) is the Fourier transform

S. Brandt and H.D. Dahmen, The Picture Book of Quantum Mechanics, 446 DOI 10.1007/978-1-4614-3951-6, © Springer Science+Business Media New York 2012 D. Wigner Distribution 447 1 +∞ i )ϕ(p) = √ exp − px ϕ(x)dx 2πh¯ −∞ h¯ of the wave function ϕ(x), i.e., )ϕ(p) is the wave function in momentum space. For the case of the Gaussian wave packet we find for the marginal distributions 1 1 (x − x )2 W (x) = √ exp − 0 , p πσ σ 2 2 x 2 x 1 1 (p − p )2 W (p) = √ exp − 0 . x σ 2 2πσp 2 p An alternative representation for the Wigner distribution can be obtained by introducing the wave function in momentum space into the expression defining W(x, p) with the help of 1 +∞ p ϕ(x) = √ exp i x )ϕ(p)dp . 2πh¯ −∞ h¯ We find +∞ 1 i q q W(x, p) = exp − xq )ϕ p − )ϕ∗ p + dq . h −∞ h¯ 2 2 A note of caution should be added: For a general wave function ϕ(x)the Wigner distribution is not a positive function everywhere. Thus, in general it cannot be interpreted as a phase-space probability density. It is, however, a real function W ∗(x, p) = W(x, p). As an example we indicate the Wigner distribution W(x, p,n)forthe eigenfunction ϕn(x) of the harmonic oscillator, (−1)n x 2 p2 1 x 2 p2 W(x, p,n) = L0 + exp − + . π n σ 2 σ 2 σ 2 σ 2 h¯ x p 2 x p

Here, the widths σx , σp are given by σx = h¯ /(2mω), σp = h¯ /(2σp), 0 = and Ln(x) is the Laguerre polynomial with upper index k 0 as discussed in Section 13.4. Figure D.1 shows the Wigner distributions for the lowest four eigenstates of the harmonic oscillator n = 0, 1, 2, 3 plotted over the plane of the scaled variables x/σx , p/σp. Accordingly, the plots are rotationally symmetric about the z axis of the coordinate frame. The nonpositive regions of W(x, p,n)can be clearly seen. The corresponding plots for the absolute square of the analyzing amplitude are shown in Figure C.6. 448 D. Wigner Distribution

Fig.D.1. Wigner distributions W(x, p,n) of the harmonic-oscillator eigenstates ϕn(x), n = 0, 1, 2, 3.

The relation to the analyzing amplitude can easily be inferred from the following observation. For an arbitrary analyzing wave function ϕD(x)we form the Wigner distribution +∞ = 1 i ϕ − y ϕ∗ + y WD(x, p) exp py D x D x dy . h −∞ h¯ 2 2

Then, the integral over x and p of the product of WD and W yields +∞ +∞ 1 +∞ 2 = ϕ∗ ϕ =| |2 WD(x, p)W(x, p)dx dp D(x) (x)dx a . −∞ −∞ h −∞

This is to say, analyzing the Wigner distribution W(x, p) of a wave function ϕ(x) with the Wigner distribution WD(x, p) of an (arbitrary) analyzing wave function ϕD(x) yields exactly the absolute square of the analyzing amplitude

+∞ = √1 ϕ∗ ϕ a D(x) (x)dx h −∞ introduced in Appendix C. D. Wigner Distribution 449

The temporal evolution of a Wigner distribution

+∞ 1 i py ∗ y y W(x, p,t) = e h¯ ψ (x + ,t)ψ(x − ,t)dy h¯ −∞ 2 2 corresponding to a time-dependent solution ψ(x,t) of the Schrödinger equation with the Hamiltonian H = p2/2m +V (x), p = (h¯ /i)d/dx, is governed by the Wigner–Moyal equation. It is the quantum-mechanical analog of the Li- ouville equation for a classical phase-space distribution. For potentials V (x) which are constant, linear, or quadratic in the coordinate x or linear combinations of these powers the two equations of Wigner and Moyal, and of Liouville are identical. For these types quantum-mechanical and classical phase-space distributions that coincide at one instant t in time, say the initial one, coincide at all the times. E. Gamma Function

The gamma function (z) introduced by Leonhard Euler is a generalization of the factorial function for integers n,

n! = 1 · 2 · 3 · ...· n ,0!= 1! = 1, to noninteger and eventually complex numbers z.ItisdeﬁnedbyEuler’s integral ∞ (z) = t z−1e−t dt ,Re(z) > 0. 0 By partial integration of

∞ t ze−t dt = (1 + z) 0 we ﬁnd the recurrence formula ∞ + =− z −t ∞ + z−1 −t = (1 z) t e 0 z t e dt z (z) 0 valid for complex z. From Euler’s integral we obtain

(1) = 1 and, thus, with the help of the recurrence relation for non-negative integer n,

(1 + n) = n!.

Euler’s integral can also be computed in closed form for z = 1/2, 1 √ = π , 2 so that – again through the recurrence relation – it is easy to ﬁnd the gamma function for positive half-integer arguments.

S. Brandt and H.D. Dahmen, The Picture Book of Quantum Mechanics, 450 DOI 10.1007/978-1-4614-3951-6, © Springer Science+Business Media New York 2012 E. Gamma Function 451

For nonpositive integer arguments the gamma function has poles as can be read off the reﬂection formula π πz (1 − z) = = . (z)sin(πz) (1 + z)sin(πz) In Figure E.1 we show graphs of the real and the imaginary part of (z) as surfaces over the complex z plane. The most striking features are the poles for nonpositive real integer values of z. For real arguments z = x the gamma function is real, i.e., Im((x)) = 0. In Figure E.2 we show (x)and1/(x). The latter function is simpler since it has no poles. The gamma function for purely imaginary arguments z = iy, y real, is shown in Figure E.3. For complex argument z = x + iy an explicit decomposition into real and imaginary part can be given, ∞ | j + x| (x + iy) = (cosθ + isinθ)|(x)| , 2 + + 2 j=0 y ( j x) where the angle θ is determined by ∞ y y θ = yψ(x) + − arctan . j + x j + x j=0 Here ψ(x)isthedigamma function d (x) ψ(x) = (ln(x)) = . dx (x) For integer n the following formula follows from the recurrence formula: (n + z) = (1 + (n − 1 + z)) = (n − 1 + z)(n − 2 + z) · ...· (1 + z)z(z). For purely imaginary z = iy we ﬁnd (n + iy) = (n − 1 + iy)(n − 2 + iy) · ...· (1 + iy)iy(iy). The gamma function of a purely imaginary argument can be obtained by spe- cialization of the argument x + iy to x = 0in(x + iy) to yield π (iy) = (sinθ − icosθ) y sinh y with ∞ y y θ =−γ y + − arctan , j j j=1 where Euler’s constant γ is given by − n 1 1 γ =−ψ(1) = lim − lnn = 0.5772156649... . n→∞ k k=1 452 E. Gamma Function

Fig.E.1. Real part (top) and imaginary part (bottom) of (z) over the complex z plane. E. Gamma Function 453

Fig.E.2. The functions (x) and 1/(x) for real arguments x. 454 E. Gamma Function

Fig.E.3. Real and imaginary parts of the gamma function for purely imaginary arguments. F. Bessel Functions and Airy Functions

Bessel’s differential equation

2 2 d Zν(x) dZν(x) 2 2 x + x + (x − ν )Zν(x) = 0 dx 2 dx is solved by the Bessel functions of the ﬁrst kind Jν(x), of the second kind (also called Neumann functions) Nν(x), and of the third kind (also called Han- (1) (2) kel functions) Hν (x)andHν (x) which are complex linear combinations of the former two. The Bessel functions of the ﬁrst kind are ∞ k x ν (−1)k x 2 Jν(x) = , 2 k!(ν + k + 1) 4 k=0 where (z) is Euler’s gamma function. The Bessel functions of the second kind are 1 Nν(x) = [Jν(x)cosνπ − J−ν(x)] . sinνπ For integer ν = n one has

n J−n(x) = (−1) Jn(x).

The modiﬁed Bessel functions are deﬁned as

∞ k x ν 1 x 2 Iν(x) = . 2 k!(ν + k + 1) 4 k=0 The Hankel functions are deﬁned by

(1) Hν (x) = Jν(x) + iNν(x),

(2) Hν (x) = Jν(x) − iNν(x). The following relations hold for the connections of the functions just discussed and the spherical Bessel, Neumann, and Hankel functions, cf. Sec- tion 10.8. The spherical Bessel functions of the ﬁrst kind are

S. Brandt and H.D. Dahmen, The Picture Book of Quantum Mechanics, 455 DOI 10.1007/978-1-4614-3951-6, © Springer Science+Business Media New York 2012 456 F. Bessel Functions and Airy Functions π j (x) = J + / (x). 2x 1 2 The spherical Bessel functions of the second kind (also called spherical Neu- mann functions)aregivenby π n (x) =− N + / (x) = (−1) j− − (x), 2x 1 2 1 and the spherical Bessel functions of the third kind (also called spherical Han- kel functions of the ﬁrst and second kind) are π (+) = + = − = (1) h (x) n (x) i j (x) i[ j (x) in (x)] i H +1/2(x), 2x π (−) (2) h (x) = n (x) − i j (x) =−i[ j (x) + in (x)] =−i H (x). 2x +1/2 In Figures F.1 and F.2 we show the functions Jν(x)andIν(x)forν = −1, −2/3, −1/3, ...,11/3. The features of these functions are simple to describe for ν ≥ 0. The functions Jν(x) oscillate around zero with an amplitude that decreases with increasing x, whereas the functions Iν(x) increase mono- tonically with x.Atx = 0weﬁndJν(0) = Iν(0) = 0forν>0. Only for ν = 0 we have J0(0) = I0(0) = 1. For ν>1thereisaregionnearx = 0inwhichthe functions essentially vanish. The size of this region increases with increasing index ν. For negative values of the index ν the functions may become very large near x = 0. Closely related to the Bessel functions are the Airy functions Ai(x)and Bi(x). They are solutions of the differential equation d2 − x f (x) = 0 dx 2 and are given by ⎧ ⎪ √ ⎪ 1 2 3/2 2 3/2 ⎨⎪ x I− / x − I / x , x > 0 3 1 3 3 1 3 3 = Ai(x) ⎪ √ , ⎪ 1 2 3/2 2 3/2 ⎩ x J− / |x| + J / |x| , x < 0 3 1 3 3 1 3 3 and ⎧ ⎪ ⎪ x 2 3/2 2 3/2 ⎨⎪ I− / x + I / x , x > 0 3 1 3 3 1 3 3 Bi(x) = . ⎪ ⎪ x 2 3/2 2 3/2 ⎩ J− / |x| − J / |x| , x < 0 3 1 3 3 1 3 3 Graphs of these functions are shown in Figure F.3. Both functions oscillate for x < 0. The wavelength of the oscillation decreases with decreasing x.For x > 0 the function Ai(x) drops fast to zero whereas Bi(x) diverges. F. Bessel Functions and Airy Functions 457

Fig.F.1. Bessel functions Jν (x). 458 F. Bessel Functions and Airy Functions

Fig.F.2. Modiﬁed Bessel functions Iν (x). F. Bessel Functions and Airy Functions 459

Fig.F.3. Airy functions Ai(x) and Bi(x). G. Poisson Distribution

In Section 3.3 we ﬁrst introduced the probability density ρ(x), which is normalized to one, +∞ ρ(x)dx = 1. −∞ We also introduced the concepts of the expectation value of x, +∞ x = xρ(x)dx , −∞ and of the variance of x, = σ 2 = − 2 var(x) x (x x ) . We now replace the continuous variable x by the discrete variable k which can assume only certain discrete values, e.g., k = 0,1,2,... . In a statistical process the variable k is assumed with the probability P(k). The total probability is normalized to one, P(k) = 1, k where the summation is performed over all possible values of k. The average value, mean value,orexpectation value of k is k = kP(k), k and the variance of k is var(k) = σ 2(k) = (k − k )2 = (k − k )2 P(k). k The simplest case is that of an alternative. The variable only takes the values κ = 0,1 . The process yields with probability p = P(1) the result κ = 1 and with probability P(0) = 1 − p the result κ = 0. Therefore, the expectation value of κ is κ =0 · (1 − p) + 1 · p = p .

S. Brandt and H.D. Dahmen, The Picture Book of Quantum Mechanics, 460 DOI 10.1007/978-1-4614-3951-6, © Springer Science+Business Media New York 2012 G. Poisson Distribution 461

We now consider a process which is a sequence of n independent alternatives each yielding the result κi = 0,1, i = 1,2,...,n. We characterize the result of the process by the variable

n k = κi , i=1 which has the range k = 0,1,...,n .

A given process yields the result k if κi = 1fork of the n alternatives and κi = 0for(n − k) alternatives. The probability for the sequence

κ1 = κ2 =·...·=κk = 1, κk+1 =·...·=κn = 0 is pk(1 − p)n−k. But this is only one particular sequence leading to the result k. In total there are n = n! k k!(n − k)! such sequences where

n! = 1 · 2 · 3 · ...· n ,0!= 1! = 1.

Therefore, the probability that our process yields the result k is n P(k) = pk(1 − p)n−k . k

This is the binomial probability distribution. The expectation value can be computed by introducing P(k) into the deﬁnition of k or, even simpler, from

n k = κi =np . i=1 In Figure G.1 we show the probabilities P(k) for various values of n but for a ﬁxed value of the product λ = np. The distribution changes drastically for small values of n but seems to approach a limiting distribution for very large n. Indeed, we can write ! λ n n! λ k 1 − P(k) = n ! k!(n − k)! n − λ k 1 n 462 G. Poisson Distribution

Fig.G.1. Binomial distributions for various values of n but ﬁxed product np = 3.

λk λ n n(n − 1) · ...· (n − k + 1) = − ! 1 k k! n nk 1 − λ ! !n ! λk λ n 1 − 1 1 − 2 · ...· 1 − k−1 = 1 − n n ! n . k! n − λ k 1 n In the limit n →∞every term in brackets in the last factor approaches one, and since λ n lim 1 − = e−λ n→∞ n we have λk P(k) = e−λ . k! This is the Poisson probability distribution. It is shown for various values of the parameter λ in Figure G.2. The expectation value of k is

∞ λk ∞ λk k = k e−λ = k e−λ k! k! k=0 k=1 ∞ ∞ λλk−1 λ j = e−λ = λ e−λ = λ . (k − 1)! j! k=1 j=0 G. Poisson Distribution 463

Fig.G.2. Poisson distributions for various values of the parameter λ.

In a similar way one ﬁnds k2 =λ(λ + 1) and therefore, also the variance of k is equal to λ, var(k) = (k − k )2 = k2 − 2k k + k 2 = k2 −2 k 2 + k 2 = k2 − k 2 = λ(λ + 1) − λ2 = λ . The Poisson distribution is markedly asymmetric for small values of λ.For large λ, however, it becomes symmetric about its mean value λ andinthat case its bell shape resembles that of the Gaussian distribution. Index

Acceleration, sudden, 78 Azimuth, 192 Airy functions, 72, 456 α particle, 86, 386, 395 Balmer series, 378 Analyzing amplitude, 430, 435 Band spectra, 133, 380 – coherent-state – coherent-state, 444 Base – coherent-state – eigenstate, 442 – states, 425 – eigenstate – coherent-state, 441 – vectors, 416 – eigenstate – eigenstate, 440 Bessel functions, 455 – free particle, 434 – modiﬁed, 455 – general case, 439 – spherical, 210, 211, 455 Binomial distribution, 461 – harmonic oscillator, 439 Bivariate Gaussian, 47 Angular Bloch equations, 404 – distribution, 205 Bloch, Felix, 400 – frequency, 11 Boersch, Hans, 6 – momentum, 191 Bohm, David, 153 – – variances, 200 Bohr – – expectation values, 199 – magneton, 397 – – in polar coordinates, 192 – radius, 268 – – interpretation of eigenfunctions, Bohr, Niels, 268 201 Born, Max, 37 – – intrinsic , 353 Bose–Einstein particles, 172 – – of free wave packet, 219 Boson, 172, 298 – – of plane waves, 214 Bound states – – vector, semiclassical, 207, 358 – and resonances, 322 ––z component, 195 –inatoms,377 Antiquarks, 389 –incrystals,377 Antisymmetry, 172 – in deep square well, 111 Argand diagram, 94, 247, 314 – in harmonic oscillator, 117 Atomic – in nuclei, 377 – beam, 399 – in periodic potential, 133 – number, 381 – in spherical square well, 251 Average value, 38, 460 – in square well, 129

S. Brandt and H.D. Dahmen, The Picture Book of Quantum Mechanics, 465 DOI 10.1007/978-1-4614-3951-6, © Springer Science+Business Media New York 2012 466 Index

– stationary, 72 Covariance – three-dimensional, 229, 251 – ellipse, 48 Boundary condition, 57, 72 – ellipsoid, 187, 265 Breit–Wigner distribution, 110 Cross section Bulk matter, 400 – differential, 241, 242, 375 – partial, 243 Carrier wave, 21, 33 – total, 243, 249 Centrifugal Crystal, 133, 380 – barrier, 226 – potential, 225 Davisson, Clinton, 6 Charm quark, 389 Deceleration, sudden, 78, 83 Chemical shift, 409 Deep square well Classical phase space – particle motion, 112 – distribution – stationary states, 111 Dehmelt, Hans, 397 – – free motion, 51 Del, 185 – – harmonic motion, 126 De Broglie – – Kepler motion, 285, 350 –wave,32 – – linear potential, 97 – wavelength, 6 – – sudden deceleration, 83 De Broglie, Louis, 6 – probability density, 51 Differential operator, 414 Classical statistical description, 50 Diffraction, 232 Classical turning point, 69 – of electrons, 6 Coherent state, 125 Digamma function, 451 Commutation relations, 191, 420, 427 Directional distribution, 202 Compton Discrete spectrum, 62 – effect, 4 Dispersion, 35 Compton, Arthur H., 6 Distinguishable particles, 157 Conduction band, 137, 380 Distribution Confluent hypergeometric function, – angular, 205, 357 344 – directional, 202, 356 Constituent wave, 16, 61 Continuity equation, 143, 145 Effective Continuous spectrum, 62 – field, 370 Contour plot, 272 – potential, 225 Correlation coefficient, 47 Eigenfunction, 56, 413 Coulomb – degenerate, 258 – potential, 267 – of angular momentum, 192 – scattering, 337 – of momentum, 186 – – phase, 344 – of spherical harmonic oscillator, 258 Coupled harmonic oscillators, 170 – orthonormality, 414 – distinguishable particles, 157 – simultaneous, 186 – indistinguishable particles, 170 Eigenstate, 56, 428 – stationary states, 167, 173 – degenerate, 258 Index 467

Eigenvalue, 56, 413, 428 Gamma function, 450 – equation, 413 γ quantum, 3 Einstein, Albert, 3 Gamow, George, 395 Elastic scattering, 58 Gaussian distribution, 22 – classical, 241 – bivariate, 47 Electric ﬁeld, 11 Geiger, Hans, 395 Electron, 2 Gerlach, Walther, 7 – spin resonance, 397, 399, 400 Germer, Lester, 6 – volt, 373 Goeppert-Mayer, Maria, 386 Electron–positron annihilation, 389 Ground state, 112 Elementary charge, 373 Group velocity, 35 Elements, Periodic Table, 382 Gyromagnetic Energy – factor, 397 – band, 137 – ratio, 359, 397 – density, 15 – ionization, 383 Hahn, Erwin, 410 – kinetic, 6, 373 Half-life, 91 – relativistic, 4, 373 Hallwachs, Wilhelm, 2 – spectrum Hamilton operator, 47 – – discrete, 74 Hamiltonian, 47, 413, 417 – – of spherical harmonic oscillator, Hankel functions, 455 257 – spherical, 211, 456 – – of the hydrogen atom, 268, 378 Harmonic oscillator Error function, 149 – analyzing amplitude, 439 Euler’s – coupled, 157 – constant, 451 – matrix representation, 419 – integral, 450 – particle motion, 119, 121, 265 Euler, Leonhard, 450 – potential, 117 Exclusion principle, 173 – spherical, 257 Expectation value, 38, 47, 424, 460 – stationary state, 116 Exponential decay, 91 Harmonic plane wave, 11 Factorial function, 450 Haxel, Otto, 386 Fermi–Dirac particles, 172 Heisenberg’s uncertainty principle, 44 Fermion, 172, 298 Heisenberg, Werner, 44 Field Hermite polynomials, 117 – electric, 11 – recurrence relations, 420 – magnetic, 11 Hermitean Fine-structure constant, 267 – conjugate, 426 Force, 58, 59 – matrix, 426 – nuclear, 375 – – operator, 420 – range of, 58 Hertz, Heinrich, 2 Free fall, 97 Hilbert space, 419 Free induction signal, 409 Hybridization, 294 – parameter, 295 468 Index

– sp, 303 Lifetime – sp2, 305 – of metastable states, 91, 395 – sp3, 306 – of radioactive nuclei, 395 Hydrogen Light – atom, 267 – quantum, 3 – spectrum, 377 –wave,11 Hyperfine structure, 400 Linear – operator, 417 Image function, 417 – superposition, 416 Impact parameter, 221 – transformation, 417 Index, refractive, 15 – vector space, 415 Indistinguishable particles, 157, 170 – classical considerations, 175 Magic numbers, 386 Intensity, 38 Magnetic Interference – field, 11 – in scattering of electrons, 6 – moment, 7 – in scattering of light, 6 – – motion in magnetic field, 359 – term, 179 – – intrinsic, 353 Intrinsic – – operator, 359 – angular momentum, 353 – quantum number, 195 – magnetic moment, 353 – resonance, 363, 366 Ionization energy, 383 – – rotating frame, 369 Magnetic-resonance experiments, 396 Jensen, Hans, 386 Magnetization, 401, 403 Joint probability density, 157 – equilibrium, 404 J/ψ particle, 391 – longitudinal, 404 – transverse, 403 Kepler motion Magneton – on elliptic orbits, 282 – Bohr, 397 – on hyperbolic orbits, 346 – nuclear, 398 Kinetic energy, 6, 373 Marginal distribution, 47 Kronecker symbol, 197, 414 Matrix – exponential, 422 Laguerre polynomial, 258, 268 – Hermitean, 420, 426 Laplace operator, 208 – mechanics, 415 – in polar coordinates, 210 – multiplication, 418 Laplace’s differential equation, 340 – operator, 418 Larmor precession, 362 Mean value, 460 Legendre Metastable state, 86, 94, 395 – functions Millikan, R. A., 2 – – associated, 195 Minimum-uncertainty state, 125 – polynomials, 195 Molecular beam, 399 – – orthogonality, 243 Momentum Lenard, Philipp, 2 – operator, 42 Index 469

– relativistic, 4 Passage through magnetic resonance, – vector, 185 406 – vector operator, 185 Pauli Moseley, Henry, 383 – equation, 360 – exclusion principle, 173, 298 Nabla, 185 – matrices, 426 Neumann functions, 455 – – vector of, 427 – spherical, 211, 456 Pauli, Wolfgang, 173 Node, 20, 74 Period, 12 – surface, 282 Periodic potential, 133 Norm, 414, 417 Periodic Table, 382 Normal oscillations, 162 Phase, 11, 33 Normalization, 37, 72 Phase-shift analysis, 314 Nuclear – in nuclear and particle physics, 391 – force, 375 Phase space – magnetic resonance, 399, 400 – analysis, 430 – magneton, 398 – probability density, classical, 51 – shell model, 386 Phase velocity, 11 Nucleon number, 386 – of de Broglie waves, 32 Nuttall, John Mitchell, 395 Photoelectric effect, 2 Operator Photon, 3 – Hamiltonian, 47, 417 Pion-proton scattering, 375, 389 – Hermitean, 420 Planck’s constant, 1 – Laplace, 208 Planck, Max, 1 – of angular momentum, 191 Plane wave, 11 – of kinetic energy, 46, 414 – angular-momentum representation, – of magnetic moment, 359 214 – of momentum, 42, 414, 417 – diffraction, 232 – – vector, 185 – three-dimensional, 185 – of position, 417 Poisson distribution, 462 – of spin, 353, 354 Polar – temporal evolution, 423 – angle, 192 Optical theorem, 249 – coordinate system, 215 Orientation axis, 295 – coordinates, 192 Orthonormal basis, 414, 425 Position Orthonormality, 199 – expectation value, 38 Overlap, 435 – quantile, 141 – vector, 185 Parity, 77, 112 Potential Partial – barrier, 85 – scattering amplitude, 233 – centrifugal, 225 – – unitarity relation, 247 – Coulomb, 267 – wave, 215 – deep square well, 111 – – scattered, 236 – double barrier, 86 470 Index

– effective, 225 Refractive index, 15 – harmonic oscillator, 117 Regge trajectory, 393 – linear, 68 Regge, Tullio, 393 – periodic, 133 Relativity, 373 – piecewise constant, 58 Relaxation – piecewise linear, 70, 131 – spin–lattice, 404 – quantum, 154 – spin–spin, 403 – repulsive shell, 325 – times, measurement, 409 – square-well, 68, 129 Repulsive shell, 325 Precession frequency, 360 Resonance, 94 Principal quantum number, 268 – of transmission, 18 Probability, 424, 460 Resonance scattering, 314 – current, 77, 143 – by a repulsive shell, 325 – – density, 145, 147 – off atoms, 387 – density, 37, 47, 460 – off molecules, 387 – – directional, 202 – off nuclei, 387 – – joint, 157 – off particles, 387 – interpretation, 37, 423 Rest – – of tunnel effect, 86 – energy, 373 – transport, 187 – mass, 373 Projectile, 374 Ripple tank, 234 Proton radius, 375 Purcell, Edward, 400 Scalar product, 415, 416 Scattered Quantile, 141, 187 – partial wave, 236 – position, 141 – wave, 234 – trajectory, 141, 149, 150 Scattering – velocity, 141 – amplitude, 236 Quantum – – partial, 233 – number, 112 – Coulomb, 337 – – magnetic, 195 – cross section, 241 – – of angular momentum, 195 – – differential, 242 – – principal, 268 – – partial, 243 – – spin, 354 – – total, 243 –ofenergy,1 – elastic, 58 – of light, 3 – potential, 154 –matrix Quarks, 389 – – element, 233, 245 – – unitarity, 245 Rabi’s formula, 368 – of atoms, 374 Rabi, Isidor, 368, 399 – of electrons, 374 Radial – of neutrons, 374 – Schrödinger equation, 210, 225 – of pions, 374 – wave function, 210, 225 – phase, 245 Radioactive nuclei, 395 – – Coulomb, 344 Index 471

– resonance, 314 Square well – three-dimensional, 226, 232 – potential, 68, 129 Schrödinger, Erwin, 46 – spherical, 251 Schrödinger equation, 46 – stationary states, 129 – for the hydrogen atom, 267 Squeezed states, 125 – in one dimension, 56 Stationary – radial, 210, 225 – bound states, 72 – stationary, 413, 428 – scattering solution, 58 – – scattering solution, 58 – – three-dimensional, 226 – three-dimensional, 208, 225 – state, 56 – time-dependent, 421, 427 –wave,15 – time-independent, 56 Statistical description, classical, 50 Schwartz’s inequality, 417 Stern, Otto, 7 Separation of variables in Schrödinger Stern–Gerlach experiment, 7 equation, 56 Suess, Hans, 386 Shell model Superposition, 20 – atomic, 380 – of degenerate eigenfunctions, 263 – nuclear, 386 Susceptibility, 405 Solid-angle element, 204 – absorptive part, 406 Sommerfeld, Arnold, 267 – dispersive part, 406 Spectral Symmetry, 172 – function, 22, 33, 187 – series, 378 Target, 374 Spectrum Temporal-evolution operator, 423 – atoms, 377 Time development, 12 – band, 133 Transition-matrix elements, 96 – continuous, 62 Transmission, 18 – deep square well, 111 – coefﬁcient, 20 – discrete, 62 – double barrier, 86 – harmonic oscillator, 116 – tunnel effect, 68 – hydrogen, 268, 377 Tunnel effect, 68, 85, 141 – square well, 129 – α particle, 395 Spherical Turning point, classical, 69 – harmonics, 192 Two-level system, 428 – square well, 251 Two-particle wave function Spin, 353 – distinguishable particles, 157 – echo, 410 – indistinguishable particles, 170 – operator, 353, 354 – quantum numbers, 354 Uncertainty, 43 – spatial probability distribution, 355 – principle, 44 – states, 353 Unitarity relation, 94, 247 Spin–lattice relaxation, 404 Units, 373 Spin–orbit interaction, 386 Υ particle, 391 Spin–spin relaxation, 403 472 Index

Valence – number, 11, 35 – band, 380 – optics, 11 – electrons, 299 – packet, 22, 33 Variance, 41, 460 – – coupled harmonic oscillator, 174 Vector ––Gaussian,23 – model, semiclassical, 207, 358 – – in harmonic oscillator, 121, 265 – space, linear, 415 – – in three dimensions, 185 Velocity ﬁeld, 189 – – partial-wave decomposition, 218 – – tunnel effect, 85 Wave – plane, 11, 185, 214 – constituent, 16, 61 – stationary, 15 – de Broglie, 32 – vector, 186 – equation, 46 Wavelength, 12 – function –deBroglie,6 – – antisymmetric, 172 Width of wave packet, 22, 47 – – distinguishable particles, 167 Wigner – – indistinguishable particles, 172 – distribution, 445, 446 – – radial, 210, 225 – functions, 202, 355 – – symmetric, 172 Wigner, Eugene, 446 – harmonic, 11 X-ray, 383 – light, 11 – mechanics, 11 Zavoisky, E., 400 Frequently Used Symbols a Bohr radius H Hamiltonian a(x0, p0, xS, pS) analyzing amplitude j probability current density

AI, AII,... amplitude factors in regions j (ρ) spherical Bessel function B , B ,... I, II, ...of space I II k wave number B magnetic-induction ﬁeld angular-momentum c speed of light quantum number c correlation coefﬁcient α Ln Laguerre polynomials dmm Wigner function L angular momentum

Dmm Wigner function Lˆ angular-momentum operator dσ/dΩ differential scattering m quantum number cross section of z component e elementary charge of angular momentum m, M mass Ec complex electric field strength E energy M(x,t) amplitude function M magnetization En energy eigenvalue n refractive index Ekin kinetic energy E electric field strength n principal quantum number ρ f (k) spectral function with respect to n ( ) spherical Neumann function wave number n unit vector f (p) spectral function with respect to p momentum momentum pˆ momentum operator f (ϑ) scattering amplitude p momentum expectation value f partial scattering amplitude p momentum vector f (Θ,Φ) directional distribution m pˆ vector operator of momentum g gyromagnetic factor P Legendre polynomial h Planck’s constant Pm associated Legendre function h¯ = h/(2π) (+) r relative coordinate h (ρ) spherical Hankel function of the first kind r radial distance (−) r position vector h (ρ) spherical Hankel function of the second kind R center-of-mass coordinate R(r), R (k,r) radial wave function η (r) scattered partial wave

Rn radial eigenfunction ϑ,Θ polar angle s spin quantum number ϑ scattering angle

S = (S1,S2,S3) spin-vector operator λ wavelength

S scattering-matrix element μ0 vacuum permeability t time μ reduced mass T oscillation period μ magnetic moment T transmission coefficient ρ probability density T kinetic energy ρcl classical phase-space probability density TT, TR transition-matrix elements σ width of ground state U voltage 0 of harmonic oscillator v0 group velocity σ1, σ2, σ3 Pauli matrices vp phase velocity σk width in wave number V potential (energy) σ partial cross section eff V effective potential σp width in momentum w average energy density σx width in position W Wigner distribution σtot total cross section W energy ϕ(x) stationary wave function W , W m coefficients in the angular ϕ stationary harmonic decomposition of a wave packet p(r) wave function x position ϕ state vector of stationary state x position expectation value φ,Φ azimuthal angle Y m spherical harmonic χ magnetic susceptibility Z atomic number χ general spin state α fine-structure constant ψ(x,t) time-dependent wave function δ scattering phase shift ψp(r,t) harmonic wave function k wave-number uncertainty ψ state vector p momentum uncertainty ω angular frequency x position uncertainty Ω solid angle ε0 vacuum permittivity ∇ nabla (or del) operator η η 1 1, −1 spin base states 2 ∇2 Laplace operator ηk(r) scattered wave Basic Equations

1 i i de Broglie wave ψ (r,t) = exp − Et exp p · r p (2πh¯ )3/2 h¯ h¯

time-dependent ∂ h¯ 2 ih¯ ψ(r,t) = − ∇2 + V (r) ψ(r,t) Schrödinger equation ∂t 2M

stationary h¯ 2 − ∇2 + V (r) ϕ (r) = Eϕ (r) Schrödinger equation 2M E E

h¯ ∂ ∂ ∂ h¯ momentum operators pˆ = (pˆ , pˆ , pˆ ) = , , = ∇ x y z i ∂x ∂y ∂z i

angular-momentum h¯ Lˆ = r × pˆ = r ×∇ operators i

radial Schrödinger h¯ 2 1 d2 ( + 1) 2M equation for spherically − r − − V (r) R (k,r) = ER (k,r) 2 2 ¯ 2 symmetric potential 2M r dr r h

ikr + · e stationary scattering wave ϕ( )(r)−−−−−→eik r + f (ϑ) k kr 1 r

∞ 1 scattering amplitude f (ϑ) = (2 + 1) f (k)P (cosϑ) k =0

σ π ∞ differential, partial, d 2 4 2 =|f (ϑ)| , σ = (2 + 1)| f (k)| , σtot = σ and total cross sections dΩ k2 =0 Physical Constants

− − Planck’s constant h = 4.136 · 10 15 eVs = 6.626 · 10 34 Js h¯ = h/(2π) = 6.582 · 10−16 eVs = 1.055 · 10−34 Js

− speed of light c = 2.998 · 108 ms 1

− elementary charge e = 1.602 · 10 19 C

e2 1 ﬁne-structure constant α = = 4πε0hc¯ 137.036

2 −31 electron mass me = 0.5110MeV/c = 9.110 · 10 kg

2 −27 proton mass mp = 938.3MeV/c = 1.673 · 10 kg

2 −27 neutron mass mn = 939.6MeV/c = 1.675 · 10 kg

Conversion Factors

− mass 1kg= 5.609 · 1035 eV/c2 ,1eV/c2 = 1.783 · 10 36 kg energy 1J= 6.241 · 1018 eV , 1eV = 1.602 · 10−19 J

− − − momentum 1kgms 1 = 1.871 · 1027 eV/c ,1eV/c = 5.345 · 10 28 kgms 1