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A. Simple Aspects of the Structure of Quantum Mechanics

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A. Simple Aspects of the Structure of Quantum Mechanics 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 op- erators. To keep things simple we shall restrict ourselves to systems with dis- crete 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 eigen- value 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 con- jugate) 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 trans- formation 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 com- bination 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 opera- tors 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 func- tion 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 momen- tum 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) .
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