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Chapter 4 Time–Independent Schrödinger Equation

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Chapter 4 Time–Independent Schrödinger Equation Chapter 4 Time{Independent Schr¨odinger Equation 4.1 Stationary States We consider again the time dependent Schr¨odingerequation (Prop. 2.1) @ 2 i (t; x) = − ~ ∆ + V (x) (t; x) = H (t; x) ; (4.1) ~ @t 2m where the potential in the Hamiltonian is assumed to be time independent V = V (x). We calculate the solutions of this equation by using the method of separation of variables, i.e. we make the following ansatz for the solution (t; x): (t; x) = (x) f(t) (4.2) and insert it into the time dependent Schr¨odingerequation, Eq. (4.1), @ f(t) 2 @2 (x) 1 i (x) = − ~ f(t) + V (x) (x)f(t) j · ~ @t 2m @x2 (x) f(t) 1 df(t) 2 1 d2 (x) i = − ~ + V (x) : (4.3) ~ f(t) dt 2m (x) dx2 Since now the left hand side in Eq. (4.3) is only dependent on t and the right hand side only on x, both sides must be equal to a constant, which we will call E, and we can thus solve each side independently. The left side yields 1 df(t) df i i~ = E ) = − E dt f(t) dt f ~ i ) ln(f) = − E t + const. ) f = const. e−i E t =~ : (4.4) ~ The constant in Eq. (4.4) will later on be absorbed into (x). 69 70 CHAPTER 4. TIME{INDEPENDENT SCHRODINGER¨ EQUATION Then multiplying the right side of Eq. (4.3) with (x) we get 2 1 d2 (x) 2 d2 (x) − ~ + V (x) = E ) − ~ + V (x) (x) = E (x) : (4.5) 2m (x) dx2 2m dx2 | {z } H (x) The operators on the left express the Hamiltonian H acting on (x), which represents the time independent Schr¨odingerequation. Theorem 4.1 (Time-independent Schr¨odingerequation) H (x) = E (x) 2 ~ where H = − 2m ∆ + V (x) is the Hamiltonian Definition 4.1 A state is called stationary, if it is represented by the wave function (t; x) = (x) e−i E t=~ . For such states the probability density is time independent j (t; x)j2 = ∗(x) (x) ei E t=~ e−i E t=~ = j (x)j2 : (4.6) | {z } 1 The expectation values of observables A(X; P ) are time independent as well Z @ h A(X; P ) i = dx ∗(x) ei E t=~ A(x; −i ) (x) e−i E t=~ ~@x Z @ = dx ∗(x) A(x; −i ) (x) : (4.7) ~@x Remark I: As a consequence, the eigenvalues of the Hamiltonian, which are the possible energy levels of the system, are clearly time independent. To see it, just take H(X; P ) instead of A(X; P ) in Eq. (4.7) and use the time- independent Schr¨odingerequation (Theorem 4.1) Z Z Z h H(X; P ) i = dx ∗(x) H (x) = dx ∗(x) E (x) = E dx ∗(x) (x) : (4.8) | {z } < 1 4.1. STATIONARY STATES 71 Remark II: The normalization of the wavefunction will restrict the possible values of the constant E, the energy of the system, in the Schr¨odingerequation. Two more interesting features about stationary states and the corresponding energies will be formulated here in the form of two lemmata, whose proofs we will leave as exercises. Lemma 4.1 For normalizable solutions (x) of the Schr¨odinger equation the energy E must be real, E 2 R. Lemma 4.2 Solutions (x) of the time-independent Schr¨odinger equation can always be chosen to be real. Definition 4.2 The parity operator P acting on a function f(x) changes the sign of its argument: P f(x) = f(−x) . We conclude that even and odd functions are eigenfunctions of the parity operator P even = + even P odd = − odd ; (4.9) which we will use in the following theorem that will be helpful later on. Theorem 4.2 For a symmetric potential V (x) = V (−x) a basis of states can be chosen, that consists entirely of even and odd functions. even(x) = (x) + (−x) odd(x) = (x) − (−x) The proof for this theorem will be left as an exercise too. 72 CHAPTER 4. TIME{INDEPENDENT SCHRODINGER¨ EQUATION 4.2 Schr¨odingerEquation as Eigenvalue Equation A subject concerning the time-independent Schr¨odingerequation we have not yet touched is its interpretation as an eigenvalue equation. Clearly, from its form we see that stationary states j i are eigenvectors/eigenfunctions of the Hamiltonian H with eigenvalues E H j i = E j i : (4.10) It implies the exact determination of the energy E. A stationary state has a precisely defined energy. Calculating the expectation value of the Hamiltonian for a stationary system just gives h H i = h j H j i = h j E j i = E h j i = E: (4.11) Consequently, there is no energy uncertainty ∆E for these states q p ∆E = ∆H = h H2 i − h H i2 = E2 − E2 = 0 : (4.12) Generally eigenvalue equations for linear operators take the form A j φ i = a j φ i ; (4.13) where a is an eigenvalue of the linear operator A with corresponding eigenvector j φ i. For hermitian operators there exist important statements about their eigenvalues and eigenfunctions. Theorem 4.3 The eigenvalues of hermitian operators are real and the eigenvectors corre- sponding to different eigenvalues are orthogonal. The proof is easy and again left as an exercise. The above theorem is vitally important for the spectrum fEng of the Hamiltonian, which is thereby guaranteed to be real H j n i = En j n i : (4.14) Using our notation j n i ≡ j n i the orthogonality and completeness relations (re- member equations (3.25) and (3.26)) can be written as X h n j m i = δnm j n i h n j = 1 : (4.15) n 4.3. EXPANSION INTO STATIONARY STATES 73 4.3 Expansion into Stationary States Using the spectral theorem (Theorem 3.1) we can then expand a given state into a com- plete orthonormal system of energy eigenstates j n i exactly as outlined in Section 3.3.1 X j i = cn j n i cn = h n j i : (4.16) n By inserting a continous CONS of position eigenstates (Eq. (3.32)) into the transition amplidute the expansion coefficients cn can be rewritten as Z Z ∗ cn = h n j i = dx h n j x i h x j i = dx n(x) (x) : (4.17) We can now extend the expansion from the time independent case to the time depen- dent one. We just remember the time dependent Schr¨odingerequation @ i (t; x) = H (t; x) ; (4.18) ~ @t with a particular solution −i En t=~ n(t; x) = n(x) e : (4.19) The general solution is then a superposition of particular solutions X −i En t=~ (t; x) = cn n(x) e : (4.20) n The expansion coefficients can easily be computed by setting t = 0 and taking the scalar product with m(x) Z Z ∗ ∗ X −i En 0=~ dx (x) (0; x) = dx (x) cn n(x) e m m | {z } n 1 Z X ∗ h m j (t = 0) i = cn dx m(x) n(x) : n | {z } δmn Thus the expansion coefficients are given by cn = h n j (t = 0) i : (4.21) Physical interpretation of the expansion coefficients: Let's consider an observable A with eigenstates n and eigenvalues an A j n i = an j n i : (4.22) If a system is in an eigenstate of this observable the expectation value (in this state) is equal to the corresponding eigenvalue h A i = h n j A j n i = an h n j n i = an : (4.23) 74 CHAPTER 4. TIME{INDEPENDENT SCHRODINGER¨ EQUATION Thus a measurement of the observable always produces the result an which implies that the uncertainty of the observable vanishes for this state ∆A = 0. Furthermore the measurement leaves the state unchanged, the system remains in the eigenstate A j n i −! j n i : (4.24) If the system, however, is in a general state j i, which is a superposition of eigenstates, the expectation value is given by the sum of all eigenvalues, weighted with the modulus squared of the expansion coefficients X X h A i = h j A j i = h cm m j A j cn n i n m X X ∗ X 2 = cm cn an h m j n i = jcnj an : n m | {z } n δmn (4.25) The expansion coefficients cn = h n j i can thus be regarded as a probability am- plitude for the transition from a state to an eigenstate n when the corresponding observable is measured. The actual transition probability is given by its modulus squared 2 jcnj { the probability for measuring the result an { which also obeys X 2 jcnj = 1 : (4.26) n So a measurement of an observable in a general state changes the state to one of the eigenstates of the observable. This process is often called the reduction or collaps of the wave function A j i −! j n i : (4.27) 4.4 Infinite Potential Well Our goal in the next sections is to calculate the energy eigenvalues and eigenfunctions for several Hamiltonians, i.e. for several potentials. Let us begin with the infinite potential well, represented by the potential V (x), as illustrated in Fig. 4.1, such that 0 for x 2 [ 0 ;L ] V (x) = (4.28) 1 else This means that the quantum object is limited to a certain region between x = 0 and x = L where it moves freely but cannot ever leave.
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