Section 6-17
Section 6-17 Summary
and −∞.) The more precise our first position measurement is, the greater the likelihood of introducing momenta quite different from zero and the more rapidly the wave packet spreads out as time passes.
In the first example, a packet located in half of a one-dimensional box, we saw ||2 oscillate back and forth, changing shape in the process so that the motion cannot be described with a single frequency. A related important case is that of an oscillator moving in a harmonic potential. Let us assume the oscillator’s position at t = 0 to be described by a gaussian wavefunction. If this function is not centered at the oscillator’s equilibrium position, we have a time dependent situation analogous to the above particle-in-a-box case. For a harmonic potential, it can be shown11 that ||2 remains a gaussian function as time passes (i.e., does not change shape), and that the center of this gaussian oscillates back-and-forth about the equilibrium position with the classical frequency. This is a situation of interest to spectroscopists because it bears on the process of electronically exciting a sample of diatomic molecules. Suppose the molecules are initially in their ground electronic and ground vibrational states. Then their vibrational wavefunction is a simple gaussian function (the lowest-energy harmonic oscillator wavefunction) centered at Re (i.e., not off-center). If the molecules are excited by a laser pulse to a new electronic state having an equilibrium internuclear distance of Re, the vibrational wavefunction at t = 0 is still the simple gaussian centered at Re, which means that it is now off-center. As time passes, the center of this function oscillates back and forth about Re in a coherent manner (i.e., describable with a single frequency). Thus, we have gone from an initial state describing an ensemble of molecules vibrating about Re with zero-point energy hv/2 and with random
phases (i.e., a time-independent state wherein such measurable properties as average molecular dipole moment appear to be constant) to a final state where the molecules are vibrating in phase about Re (a time-dependent state, wherein one might expect to see time variation of such properties). If Re were quite a bit smaller than Re, for instance, then almost all the molecules would find themselves to be “too short” at t = 0, “too long” a short time later, etc., as they swing in phase about Re. As a result of this simple behavior, it is possible to take advantage of it and time a second laser pulse to strike the molecules when they are almost all at their shortest, or all at their longest, extension.
Of course, in real molecules the potential is not exactly harmonic. Furthermore, phase coherence is eventually lost due to collisions. So the second pulse must come very soon after the first one.
6-17 Summary
Some of the postulates and proofs described in this chapter are most important for what follows in this book, and we list these points here.
1. ψ describes a state as completely as possible and must meet certain mathemati cal requirements (single-valued, etc.). ψ∗ψ is the probability density distribution function for the system.
2. For any observable variable, there is an operator (hermitian) which is constructed from the classical expression according to a simple recipe. (Operators related to 11See Schiff [3, pp. 67, 68], and Tanner [4].
Chapter 6 Postulates and Theorems of Quantum Mechanics “spin” are the exception because the classical analog does not exist.) The eigenvalues for such an operator are the possible values we can measure for that quantity. The act of measuring the quantity forces the system into a state described by an eigenfunction of the operator. Once in that state, we may know exact values for other quantities only if their operators commute with the operator associated with our measurement.
3. If the hamiltonian operator for a system is time independent, stationary eigenfunc tions exist of the form ψ(q, ω) exp(−iEt/¯h). The time-dependent exponential does not affect the measurable properties of a system in this state and is almost always completely ignored in any time-independent problem.
4. The formula for the quantum-mechanical average value [Eq. (6-9)] is equivalent to the arithmetic average of all the possible measured values of a property times their frequency of occurrence [Eq. (6-33)]. This means that it is impossible to devise a function that satisfies the general conditions on ψ and leads to an average energy lower than the lowest eigenvalue of ˆ H .
5. The square-integrable eigenfunctions for an operator corresponding to an observable quantity form a complete set, which may be assumed orthonormal. The eigenvalues are all real.
6. Any operation that leaves ˆ H unchanged also commutes with ˆ H .
7. Wavefunctions describing time-dependent states are solutions to Schr¨odinger’s time dependent equation. The absolute square of such a wavefunction gives a particle distribution function that depends on time. The time evolution of this particle distribution function is the quantum-mechanical equivalent of the classical concept of a trajectory. It is often convenient to express the time-dependent wave packet as a linear combination of eigenfunctions of the time-independent hamiltonian multiplied by their time-dependent phase factors.
6-17.A Problems6-1. Prove that d2/dx2 is hermitian.
6-2. Integrate the expressions in Eqs. (6-13) and (6-15) to show that their integrals are equal.
6-3. Prove that, if a normalized function is expanded in terms of an orthonormal set of functions, the sum of the absolute squares of the expansion coefficients is unity.
6-4. Show that a particular coefficient ck in Eq. (6-30) is given by ck = µ∗ψ dv.
k
√
6-5. A particle in a ring is in a state with wavefunction ψ = 1/ π cos(2φ).
a) Calculate the average value for the angular momentum by evaluating ψ∗ ˆLzψ dφ, where ˆLz =(¯h/i)d/dφ. (Use symmetry arguments to evaluate the integral.)
b) Express ψ as a linear combination of exponentials and evaluate the average value of the angular momentum using the formula Lz,av =
c∗c
i i i Lzi where Lzi is the eigenvalue for the ith exponential function.
Section 6-17 Summary6-6. Using Eq. (6-37), show that x · px ≥ ¯h/2.
6-7. What condition must the function φ satisfy for the equality part of ≥ to hold in Eq. (6-34)?
6-8. Suppose you had an operator and a set of eigenfunctions for it that were associated with real eigenvalues. Does it necessarily follow that the operator is hermitian as defined by Eq. (6-10)? [Hint: Consider d/dr and the set of all functions exp(−ar), where a is real and positive definite.]
6-9. a) Show that the nonstationary state having wavefunction
√
√
= (1/ 2)ψ1s exp(it/2) + (1/ 2)ψ2p exp(it/8)
0
is a solution to Schr¨odinger’s time-dependent equation when Hˆ is the timeindependent ˆ H of the hydrogen atom. Use atomic units (i.e., ¯h = 1).
b) This time-dependent state is dipolar and oscillates with a characteristic fre quency ν. Show that ν satisfies the relation E2 − E1 ≡ E = 2πν in a.u.
(The dipole oscillates at the same frequency as that of light required to drive the 1s ←→ 2p transition. This is central to the subject of spectroscopy.)
6-10. From the definition that φ = φ − Sψ [see the discussion following Eq. (6-26)], evaluate the normalizing constant for φ, assuming that φ and ψ are normalized.
√ 6-11. Given the two normalized nonorthogonal functions (1/ π ) exp(−r) and √1/3πr exp(−r), construct a new function φ that is orthogonal to the first function and lies within the function space spanned by these two functions, and
is normalized.
6-12. If the hydrogen atom 1s AO is expanded in terms of the He+ AOs, what is the coefficient for (a) the He+ 1s AO? (b) the He+ 2p0 AO?
6-13. The lowest-energy eigenfunction for the one-dimensional harmonic oscillator is ψn=0 = (β/π)1/4 exp(−βx2/2).
a) Demonstrate whether or not momentum is a constant of motion (i.e., is a “sharp” quantity) for this state.
b) Calculate the average momentum for this state.
6-14. Demonstrate whether x2d/dx and xd2/dx2 commute. What about xd/dx and x2d2/dx2?
6-15. Evaluate the following integrals over all space. In neither case should you need to do this by brute force.
a) (3dxy) ˆ L2(3dxy) dv
b)
3d
ˆ
xy Lz (3dxy ) d v Chapter 6 Postulates and Theorems of Quantum Mechanics6-16. The operators for energy and angular momentum for an electron constrained to move in a ring of constant potential are, respectively, in a.u. −(1/2)d2/dφ2 and (1/ i)d/dφ.
a) Discuss whether or not there should be a set of functions that are simultane ously eigenfunctions for both operators.
b) Discuss whether or not there is a set of functions that are eigenfunctions for one of these operators but not the other.
c) Discuss whether it is reasonable to expect these two physical quantities to be exactly knowable simultaneously or whether the uncertainty principle makes this impossible.
√
6-17. Suppose a hydrogen atom state was approximated by the function φ = (1/ 3)1s
√
√
+(1/ 3)2s + (1/ 3)3s, where 1s, 2s, and 3s are normalized eigenfunctions for the hydrogen atom hamiltonian. What would be the average value of energy associated with this function, in a.u.?
6-18. A function f is defined as follows: f = 0.1 · 1s + 0.2 · 2p1 + 0.3 · 3d2, where 1s is the normalized eigenfunction for the 1s state of the hydrogen atom, etc.
Evaluate the average value of the z component of angular momentum in a.u. for this function.
6-19. Without looking back at the text, prove that (a) eigenvalues of hermitian operators are real, (b) nondegenerate eigenfunctions of hermitian operators are orthogonal, (c) nondegenerate eigenfunctions of ˆ A must be eigenfunctions of ˆ
B if ˆ
A and ˆ
B
commute.
6-20. Using Eq. (6-38), obtain an expression for ||2 as a function of x and t.
6-21. Evaluate t of Fig. 6-2 in terms of m, h, and L.
6-22. Verify the values of the coefficients in Eq. (6-40). How can you tell from simple inspection that (a) c2 will be largest and positive, (b) c4 will be zero, (c) ci will tend toward small values at large i?
6-23. a) Evaluate the first two coefficients in Eq. (6-42).
b) What qualitative difference would you expect between of Eq. (6-42) and one that takes explicit account of the changing potential resulting as the beta particle travels away from the nucleus?
6-24. Show by qualitative arguments based on mathematical functions why coefficients ck should drop off more rapidly with k if the position wave packet is broader.
√
c
4
k = 2α/π exp(−αx2) exp(ikx) dx.
6-25. a) Prove that, if V is real and (x, y, z, t) satisfies Schr¨odinger’s time-dependent equation, then (x, y, z, −t)∗ is also a solution. (This is called “invariance under time reversal.”)
b) Show that, for stationary states, invariance under time reversal means that
ˆ
H ψ∗ = Eψ∗ if ˆ H ψ = Eψ and if V is real.
c) Show from (b) that nondegenerate eigenfunctions of ˆ H (with real V ) must be real.
Section 6-17 Summary d) What becomes of the 2p−1 AO (with time dependence included) upon complex conjugation and time reversal? the 2p0 AO?
e) Can the statement in (c) be reworded to say that all degenerate eigenfunctions of ˆ H (with real V ) must be complex?
Multiple Choice Questions (Try to answer these without referring to the text.)
1. Which one of the following statements about the eigenfunctions of a time independent Hamiltonian operator is true?
a) Any linear combination of these eigenfunctions is also an eigenfunction for ˆ H .
b) The state function for this system must be one of these eigenfunctions.
c) The eigenvalues associated with these eigenfunctions must all be real.
d) These eigenfunctions must all be orthogonal to one another.
e) These eigenfunctions have no time dependence.
2. A hydrogen atom is in a nonstationary state having the wavefunction 1 {ψ
2
1s
exp(it/2 ¯h) + ψ2s exp(it/8¯h) + ψ2p exp(it/8¯h) + ψ exp(it/8 ¯h)}
0
2p+1
Which statement is true at any time t?
a) A measurement of the energy has a 25% chance of giving −0.5 a.u.
b) The average value of the z component of angular momentum is 0.5 a.u.
c) The average energy is − 7 a.u.
8
d) A measurement of the total angular momentum has a 75% chance of giving √2 a.u.
e) None of the above statements is true.
3. The function rexp(−0.3r2) cos θ is expanded in terms of hydrogen atom wavefunc tions. This series may have finite contributions from bound-state eigenfunctions a) of all types: s, px, py, pz, dxy, dyz, etc.
b) of all types except s.
c) of types px, py, pz only.
d) of type pz only.
e) of types pz and dz2 only.
References [1] E. C. Kemble, The Fundamental Principles of Quantum Mechanics with Elemen-tary Applications. Dover, New York, 1958.
[2] E. Merzbacher, Quantum Mechanics, 3rd ed. Wiley, New York, 1998.
[3] L. I. Schiff, Quantum Mechanics, 3rd ed. McGraw-Hill, New York, 1968.
[4] J. J. Tanner, J. Chem. Ed., 67, 917 (1990).
and −∞.) The more precise our first position measurement is, the greater the likelihood of introducing momenta quite different from zero and the more rapidly the wave packet spreads out as time passes.
In the first example, a packet located in half of a one-dimensional box, we saw ||2 oscillate back and forth, changing shape in the process so that the motion cannot be described with a single frequency. A related important case is that of an oscillator moving in a harmonic potential. Let us assume the oscillator’s position at t = 0 to be described by a gaussian wavefunction. If this function is not centered at the oscillator’s equilibrium position, we have a time dependent situation analogous to the above particle-in-a-box case. For a harmonic potential, it can be shown11 that ||2 remains a gaussian function as time passes (i.e., does not change shape), and that the center of this gaussian oscillates back-and-forth about the equilibrium position with the classical frequency. This is a situation of interest to spectroscopists because it bears on the process of electronically exciting a sample of diatomic molecules. Suppose the molecules are initially in their ground electronic and ground vibrational states. Then their vibrational wavefunction is a simple gaussian function (the lowest-energy harmonic oscillator wavefunction) centered at Re (i.e., not off-center). If the molecules are excited by a laser pulse to a new electronic state having an equilibrium internuclear distance of Re, the vibrational wavefunction at t = 0 is still the simple gaussian centered at Re, which means that it is now off-center. As time passes, the center of this function oscillates back and forth about Re in a coherent manner (i.e., describable with a single frequency). Thus, we have gone from an initial state describing an ensemble of molecules vibrating about Re with zero-point energy hv/2 and with random
phases (i.e., a time-independent state wherein such measurable properties as average molecular dipole moment appear to be constant) to a final state where the molecules are vibrating in phase about Re (a time-dependent state, wherein one might expect to see time variation of such properties). If Re were quite a bit smaller than Re, for instance, then almost all the molecules would find themselves to be “too short” at t = 0, “too long” a short time later, etc., as they swing in phase about Re. As a result of this simple behavior, it is possible to take advantage of it and time a second laser pulse to strike the molecules when they are almost all at their shortest, or all at their longest, extension.
Of course, in real molecules the potential is not exactly harmonic. Furthermore, phase coherence is eventually lost due to collisions. So the second pulse must come very soon after the first one.
6-17 Summary
Some of the postulates and proofs described in this chapter are most important for what follows in this book, and we list these points here.
1. ψ describes a state as completely as possible and must meet certain mathemati cal requirements (single-valued, etc.). ψ∗ψ is the probability density distribution function for the system.
2. For any observable variable, there is an operator (hermitian) which is constructed from the classical expression according to a simple recipe. (Operators related to 11See Schiff [3, pp. 67, 68], and Tanner [4].
Chapter 6 Postulates and Theorems of Quantum Mechanics “spin” are the exception because the classical analog does not exist.) The eigenvalues for such an operator are the possible values we can measure for that quantity. The act of measuring the quantity forces the system into a state described by an eigenfunction of the operator. Once in that state, we may know exact values for other quantities only if their operators commute with the operator associated with our measurement.
3. If the hamiltonian operator for a system is time independent, stationary eigenfunc tions exist of the form ψ(q, ω) exp(−iEt/¯h). The time-dependent exponential does not affect the measurable properties of a system in this state and is almost always completely ignored in any time-independent problem.
4. The formula for the quantum-mechanical average value [Eq. (6-9)] is equivalent to the arithmetic average of all the possible measured values of a property times their frequency of occurrence [Eq. (6-33)]. This means that it is impossible to devise a function that satisfies the general conditions on ψ and leads to an average energy lower than the lowest eigenvalue of ˆ H .
5. The square-integrable eigenfunctions for an operator corresponding to an observable quantity form a complete set, which may be assumed orthonormal. The eigenvalues are all real.
6. Any operation that leaves ˆ H unchanged also commutes with ˆ H .
7. Wavefunctions describing time-dependent states are solutions to Schr¨odinger’s time dependent equation. The absolute square of such a wavefunction gives a particle distribution function that depends on time. The time evolution of this particle distribution function is the quantum-mechanical equivalent of the classical concept of a trajectory. It is often convenient to express the time-dependent wave packet as a linear combination of eigenfunctions of the time-independent hamiltonian multiplied by their time-dependent phase factors.
6-17.A Problems6-1. Prove that d2/dx2 is hermitian.
6-2. Integrate the expressions in Eqs. (6-13) and (6-15) to show that their integrals are equal.
6-3. Prove that, if a normalized function is expanded in terms of an orthonormal set of functions, the sum of the absolute squares of the expansion coefficients is unity.
6-4. Show that a particular coefficient ck in Eq. (6-30) is given by ck = µ∗ψ dv.
k
√
6-5. A particle in a ring is in a state with wavefunction ψ = 1/ π cos(2φ).
a) Calculate the average value for the angular momentum by evaluating ψ∗ ˆLzψ dφ, where ˆLz =(¯h/i)d/dφ. (Use symmetry arguments to evaluate the integral.)
b) Express ψ as a linear combination of exponentials and evaluate the average value of the angular momentum using the formula Lz,av =
c∗c
i i i Lzi where Lzi is the eigenvalue for the ith exponential function.
Section 6-17 Summary6-6. Using Eq. (6-37), show that x · px ≥ ¯h/2.
6-7. What condition must the function φ satisfy for the equality part of ≥ to hold in Eq. (6-34)?
6-8. Suppose you had an operator and a set of eigenfunctions for it that were associated with real eigenvalues. Does it necessarily follow that the operator is hermitian as defined by Eq. (6-10)? [Hint: Consider d/dr and the set of all functions exp(−ar), where a is real and positive definite.]
6-9. a) Show that the nonstationary state having wavefunction
√
√
= (1/ 2)ψ1s exp(it/2) + (1/ 2)ψ2p exp(it/8)
0
is a solution to Schr¨odinger’s time-dependent equation when Hˆ is the timeindependent ˆ H of the hydrogen atom. Use atomic units (i.e., ¯h = 1).
b) This time-dependent state is dipolar and oscillates with a characteristic fre quency ν. Show that ν satisfies the relation E2 − E1 ≡ E = 2πν in a.u.
(The dipole oscillates at the same frequency as that of light required to drive the 1s ←→ 2p transition. This is central to the subject of spectroscopy.)
6-10. From the definition that φ = φ − Sψ [see the discussion following Eq. (6-26)], evaluate the normalizing constant for φ, assuming that φ and ψ are normalized.
√ 6-11. Given the two normalized nonorthogonal functions (1/ π ) exp(−r) and √1/3πr exp(−r), construct a new function φ that is orthogonal to the first function and lies within the function space spanned by these two functions, and
is normalized.
6-12. If the hydrogen atom 1s AO is expanded in terms of the He+ AOs, what is the coefficient for (a) the He+ 1s AO? (b) the He+ 2p0 AO?
6-13. The lowest-energy eigenfunction for the one-dimensional harmonic oscillator is ψn=0 = (β/π)1/4 exp(−βx2/2).
a) Demonstrate whether or not momentum is a constant of motion (i.e., is a “sharp” quantity) for this state.
b) Calculate the average momentum for this state.
6-14. Demonstrate whether x2d/dx and xd2/dx2 commute. What about xd/dx and x2d2/dx2?
6-15. Evaluate the following integrals over all space. In neither case should you need to do this by brute force.
a) (3dxy) ˆ L2(3dxy) dv
b)
3d
ˆ
xy Lz (3dxy ) d v Chapter 6 Postulates and Theorems of Quantum Mechanics6-16. The operators for energy and angular momentum for an electron constrained to move in a ring of constant potential are, respectively, in a.u. −(1/2)d2/dφ2 and (1/ i)d/dφ.
a) Discuss whether or not there should be a set of functions that are simultane ously eigenfunctions for both operators.
b) Discuss whether or not there is a set of functions that are eigenfunctions for one of these operators but not the other.
c) Discuss whether it is reasonable to expect these two physical quantities to be exactly knowable simultaneously or whether the uncertainty principle makes this impossible.
√
6-17. Suppose a hydrogen atom state was approximated by the function φ = (1/ 3)1s
√
√
+(1/ 3)2s + (1/ 3)3s, where 1s, 2s, and 3s are normalized eigenfunctions for the hydrogen atom hamiltonian. What would be the average value of energy associated with this function, in a.u.?
6-18. A function f is defined as follows: f = 0.1 · 1s + 0.2 · 2p1 + 0.3 · 3d2, where 1s is the normalized eigenfunction for the 1s state of the hydrogen atom, etc.
Evaluate the average value of the z component of angular momentum in a.u. for this function.
6-19. Without looking back at the text, prove that (a) eigenvalues of hermitian operators are real, (b) nondegenerate eigenfunctions of hermitian operators are orthogonal, (c) nondegenerate eigenfunctions of ˆ A must be eigenfunctions of ˆ
B if ˆ
A and ˆ
B
commute.
6-20. Using Eq. (6-38), obtain an expression for ||2 as a function of x and t.
6-21. Evaluate t of Fig. 6-2 in terms of m, h, and L.
6-22. Verify the values of the coefficients in Eq. (6-40). How can you tell from simple inspection that (a) c2 will be largest and positive, (b) c4 will be zero, (c) ci will tend toward small values at large i?
6-23. a) Evaluate the first two coefficients in Eq. (6-42).
b) What qualitative difference would you expect between of Eq. (6-42) and one that takes explicit account of the changing potential resulting as the beta particle travels away from the nucleus?
6-24. Show by qualitative arguments based on mathematical functions why coefficients ck should drop off more rapidly with k if the position wave packet is broader.
√
c
4
k = 2α/π exp(−αx2) exp(ikx) dx.
6-25. a) Prove that, if V is real and (x, y, z, t) satisfies Schr¨odinger’s time-dependent equation, then (x, y, z, −t)∗ is also a solution. (This is called “invariance under time reversal.”)
b) Show that, for stationary states, invariance under time reversal means that
ˆ
H ψ∗ = Eψ∗ if ˆ H ψ = Eψ and if V is real.
c) Show from (b) that nondegenerate eigenfunctions of ˆ H (with real V ) must be real.
Section 6-17 Summary d) What becomes of the 2p−1 AO (with time dependence included) upon complex conjugation and time reversal? the 2p0 AO?
e) Can the statement in (c) be reworded to say that all degenerate eigenfunctions of ˆ H (with real V ) must be complex?
Multiple Choice Questions (Try to answer these without referring to the text.)
1. Which one of the following statements about the eigenfunctions of a time independent Hamiltonian operator is true?
a) Any linear combination of these eigenfunctions is also an eigenfunction for ˆ H .
b) The state function for this system must be one of these eigenfunctions.
c) The eigenvalues associated with these eigenfunctions must all be real.
d) These eigenfunctions must all be orthogonal to one another.
e) These eigenfunctions have no time dependence.
2. A hydrogen atom is in a nonstationary state having the wavefunction 1 {ψ
2
1s
exp(it/2 ¯h) + ψ2s exp(it/8¯h) + ψ2p exp(it/8¯h) + ψ exp(it/8 ¯h)}
0
2p+1
Which statement is true at any time t?
a) A measurement of the energy has a 25% chance of giving −0.5 a.u.
b) The average value of the z component of angular momentum is 0.5 a.u.
c) The average energy is − 7 a.u.
8
d) A measurement of the total angular momentum has a 75% chance of giving √2 a.u.
e) None of the above statements is true.
3. The function rexp(−0.3r2) cos θ is expanded in terms of hydrogen atom wavefunc tions. This series may have finite contributions from bound-state eigenfunctions a) of all types: s, px, py, pz, dxy, dyz, etc.
b) of all types except s.
c) of types px, py, pz only.
d) of type pz only.
e) of types pz and dz2 only.
References [1] E. C. Kemble, The Fundamental Principles of Quantum Mechanics with Elemen-tary Applications. Dover, New York, 1958.
[2] E. Merzbacher, Quantum Mechanics, 3rd ed. Wiley, New York, 1998.
[3] L. I. Schiff, Quantum Mechanics, 3rd ed. McGraw-Hill, New York, 1968.
[4] J. J. Tanner, J. Chem. Ed., 67, 917 (1990).