The Physics of Quantum Mechanics
Instructor’s Solutions Manual (James Binney) – Complete Worked
Solutions
,Problems 1
Problems
1.1 What physical phenomenon requires us to work with probability amplitudes rather than just
with probabilities, as in other fields of endeavour?
Soln: Quantum interference.
1.2 What properties cause complete sets of amplitudes to constitute the elements of a vector space?
Soln: Meaning can be given to (a) adding two sets of amplitudes and (b) multiplying a set by an
arbitrary complex number.
1.3 V ′ is the dual space of the vector space V . For a mathematician, what objects comprise V ′?
Soln: The linear functions on V .
1.4 In quantum mechanics, what objects are the members of the vector space V ? Give an example
for the case of quantum mechanics of a member of the dual space V ′ and explain how members of
V ′ enable us to predict the outcomes of experiments.
Soln: Any complete set of amplitudes is a member of V . Since a complete set of amplitudes
uniquely specifies the physical state of the system, we can consider that physical states are the
members of V . V ′ is inhabited by bras. A bra is a function that extracts from a ket the amplitude
for a specific event. For example⟨ E2| extracts from | ψ⟩ the amplitude ⟨ E2 |ψ⟩ that a measurement
of energy determines that the system is in its second excited state.
1.5 Given that |ψ⟩ = eiπ/5|a⟩ + eiπ/4|b⟩, express ⟨ψ| as a linear combination of ⟨a| and ⟨b|.
Soln:
⟨ψ| = e−iπ/5⟨a| + e−iπ/4⟨b|
1.6 What properties characterise the bra ⟨a| that is associated with the ket |a⟩?
Soln: If we choose any orthonormal basis that includes |a⟩, then ⟨a| is the linear function that
vanishes on every basis function except |a⟩.
1.7 An electron can be in one of two potential wells that are so close that it can ‘tunnel’ from one
to the other (see §5.2 for a description of quantum-mechanical tunnelling). Its state vector can be
written
|ψ⟩ = a|A⟩ + b|B⟩, (1.1)
where |A⟩ is the state of being in the first well and |B⟩ is the state of being in the second well and all
kets are correctly normalised. What is the p√robability of finding the particle in the first well given
that: (a) a = i/2; (b) b = eiπ; (c) b = 13 + i/ 2?
Soln: (a) P = 1 ; (b) P = 0; (c) P = 7 .
4 18
1.8 An electron can ‘tunnel’ between potential wells that form a chain, so its state vector can be
written
∞
Σ
|ψ⟩ = an|n⟩, (1.2a)
−∞
where |n⟩ is the state of being in the n well, where n increases from left to right. Let
th
|n|/2
1 −i
an = √ einπ . (1.2b)
2 3
a. What is the probability of finding the electron in the nth well?
b. What is the probability of finding the electron in well 0 or anywhere to the right of it?
Σ ∞
Soln: (a) P = 21 3−|n|. (b) n =1 3−n = 21 and P0 = 12 so P (n ≥ 0) = 34.
,2 Problems
Problems
2.1 How is a wavefunction ψ(x) written in Dirac’s notation? What’s the physical significance of
the complex number ψ(x) for given x?
Soln: ψ(x) ≡ ⟨x|ψ⟩. ψ(x) is the amplitude to be found at x.
⟨ Q
2.2 Let Q be an operator. Under what circumstances is the complex number a | b| ⟩equal to the
complex number (⟨ b| Q|a⟩ )∗ for any states | a⟩ and | b⟩ ?
Soln: When Q is Hermitian.
2.3 Let Q be the operator of an observable and let ψ | ⟩ be the state of our system.
a. What are the physical interpretations of ⟨ ψ| Q|ψ⟩ and |⟨qn |ψ⟩|2, where |qn ⟩is the nth eigenket
of the observable Q and qn is the corresponding eigenvalue?
Σ
b. What is the Σ operator n |qn⟩⟨qn|, where the sum is over all eigenkets of Q? What is the
operator n qn|qn⟩⟨qn|?
c. If un(x) is the wavefunction of the state |qn⟩, write down an integral that evaluates to ⟨qn|ψ⟩.
Soln: (a) ⟨ψ|Q|ψ⟩ is the e xp ecΣ tation value of Q. |⟨qn|ψ⟩|2 is the probability of∫obtaining qn on
measuring the observable Q. (b) n |qn⟩⟨qn| is the identity operator. (c) ⟨qn|ψ⟩ = dx u∗n (x)ψ(x).
2.4 What does it mean to say that two operators commute? What is the significance of two
observables having mutually commuting operators?
Given that the commutator [P, Q] = 0 for some observables P and Q, does it follow that for all
|ψ⟩ /= 0 we have [P, Q]| ψ⟩ /= 0?
Soln: Commutation implies you always get the same result no matter in which order you use the
operators. If observables have commuting operators, there is a complete set of states in which both
observables have well-defined values.
No, [P, Q] /= 0 implies only that there is some state |φ⟩ such that PQ|φ⟩ /= QP |φ⟩; one may
readily construct an example of operators such that PQ|i⟩ = QP |i⟩ for basis vectors |i⟩ with i = 2, ∞
but PQ|0⟩ = / QP |0⟩ and PQ|1⟩ /= QP |1⟩.
2.5 Let ψ(x, t) be the correctly normalised wavefunction of a particle of mass m and potential
energy V (x). Write down expressions for the expectation values of (a) x; (b) x2; (c) the momentum
px; (d) p2x; (e) the energy.
What is the probability that the particle will be found in the interval (x1, x2)?
∫∞ ∫∞
Soln: (a) ⟨x⟩ = −∞ dxx|ψ|2; (b) ⟨x2⟩ = −∞ dxx2|ψ|2;
∫∞ ∫ ∞
∫∞
dψ dxψ∗ −h̄ d ψ + V (x)ψ .
2 2 2
(c) ⟨p⟩ = − i h̄ dxψ ∗ ; (d) ⟨p2 ⟩ = −h̄ 2 dxψ ∗ d ψ ; (e) ⟨E⟩ =
dx −∞ dx2 −∞ 2m dx2
∫ x2−∞
P (x 1, x 2) = x dx |ψ| .2
1
2.6 Write down the time-independent (TISE) and the time-dependent (TDSE) Schrödinger equa-
tions. Is it necessary for the wavefunction of a system to satisfy the TDSE? Under what circumstances
does the wavefunction of a system satisfy the TISE?
Soln:
∂|ψ⟩
TISE : H|En⟩ = En|En⟩; TDSE : i h̄ = H|ψ⟩.
∂t
The state of every system must satisfy the TDSE, whereas the TISE is satisfied only if the state has
well defined energy (which for real systems is never the case).
2.7 Why is the TDSE first-order in time, rather than second-order like Newton’s equations of
motion?
Soln: Because ψ| is ⟩ a complete set of information, so for physics to be possible it must contain
complete initial conditions for solution of the time-evolution equation. If the TDSE were second-order
in t, this condition would be violated.
2.8 A particle is confined in a potential well such that its allowed energies are En = n2E , where
n = 1, 2, . . . is an integer and E a positive constant. The corresponding energy eigenstates are |1⟩, |2⟩,
. . . , |n⟩, . . . At t = 0 the particle is in the state
|ψ(0)⟩ = 0.2|1⟩ + 0.3|2⟩ + 0.4|3⟩ + 0.843|4⟩. (2.1)
a. What is the probability, if the energy is measured at t = 0, of finding a number smaller than
6E ?
, Problems 3
b. What is the mean value and what is the rms deviation of the energy of the particle in the state
|ψ(0) ⟩?
c. Calculate the state vector |ψ⟩ at time t. Do the results found in (a) and (b) for time t remain
valid for arbitrary time t?
d. When the energy is measured it turns out to be 16 E. After the measurement, what is the state
of the system? What result is obtained if the energy is measured again?
Soln: (a) 0.13; (b) the probabilities are P1 = 0.04, P2 = 0.09, P3 = 0.16, P4 = 0.71 so ⟨E⟩ = 13.2E,
⟨E2⟩ = 196E2, and σE = 4.67E. (c) |ψ(t)⟩ = 0.2e−i E t/h̄ |1⟩ + 0.3e−i4E t/h̄ |2⟩ + 0.4e−i9E t/h̄ |3⟩ +
0.843e−i16E t/h̄ |4⟩. (d) Thereafter |ψ⟩ = |4⟩ and E = 16E with certainty.
2.9 A system has a time-independent Hamiltonian that has spectrum{ En} . Prove that the prob-
ability Pk that a measurement of energy will yield the value Ek is is time-independent. Hint: you
can do this either from Ehrenfest’s theorem, or by differentiating⟨ Ek, t| ψ⟩ w.r.t. t and using the
TDSE.
Soln: This follows from Ehrenfest’s theorem: the expectation value of any function of H is con-
stant, so the probability distribution defined by the Pk is time-independent. For a more direct
answer, consider the amplitude ak(t) to get Ek: ak = ⟨Ek, t|ψ⟩, so using the TDSE and its adjoint
∂ak ∂⟨Ek, t| ∂|ψ⟩ 1
= |ψ⟩ + ⟨E k, t| = (−⟨E k, t|H|ψ⟩ + ⟨E k, t|H|ψ⟩) = 0.
∂t ∂t ∂t ih̄
2.10 Let ψ(x) be a properly normalised wavefunction and Q an operator on wavefunctions. Let
{qr } be the spectrum of Q and u{r(x) be} the corresponding correctly normalised eigenfunctions.
Write down an expression for the probability that a measurement of Q will yield the value qr. Show
Σ ∫ −∞ ψ ∗ Q̂ψ dx.1
that
Soln: r P (qr|ψ) = 1. Show further that the expectation of Q is ⟨Q⟩ ≡ ∞
∫ 2
Pi = dxui∗(x)ψ(x) where Qui(x) = qiui(x).
! ,
∫ ∫ Σ Σ Σ ∫ Σ Σ
1 = dx |ψ|2 = dx ai∗ui∗ ajuj = ai∗aj dxui∗uj = |ai|2 = Pi
i j ij i i
Similarly
! ,
∫ ∫ Σ Σ Σ ∫
dxψ∗Qψ = dx ai∗ui∗ Q ajuj = ai∗aj dxui∗Quj
i j ij
Σ ∫ Σ Σ
= ai∗ajqj dxui∗uj = |ai|2qi = Piqi
ij i i
which is by definition the expectation value of Q.
2.11 Find the energy of neutron, electron and electromagnetic waves of wavelength 0.1 nm.
Soln:
p2 (h/λ)2 mn
En = = = 1.32 × 10−20 J = 0.0821 eV; E e=
m E = 1837En = 150.8 eV;
2m 2m e n
Eγ = hν = hc/λ = 1.988 × 10−15 J = 12.407 keV.
2.12 Neutrons are emitted from an atomic pile with a Maxwellian distribution of velocities for
temperature 400 K. Find the most probable de Broglie wavelength in the beam.
Soln: The rate at which a neutron hits the wall of the pile is proportional to its momentum p, so the
2 2
probability density of neutrons hitting the wall is proportional to p3e−p /2mkT = (2mkT )3/2x3e−x ,
√
√ p = 2mkT x.√The probability density is extremised when (3x2 − 2x4)e−x = 0, √
2
where i.e., when
x = 3/2 and p = 3mkT . Thus the most probable de Broglie wavelength is λ = h/ 3mkT =
0.126 nm.
1 In an elegant formulation of quantum mechanics, this last result is the basic postulate of the theory, and one
derives other rules for the physical interpretation of the qn, an, etc., from it – see J. von Neumann, Mathematical
Foundations of Quantum Mechanics, Princeton University Press.
Instructor’s Solutions Manual (James Binney) – Complete Worked
Solutions
,Problems 1
Problems
1.1 What physical phenomenon requires us to work with probability amplitudes rather than just
with probabilities, as in other fields of endeavour?
Soln: Quantum interference.
1.2 What properties cause complete sets of amplitudes to constitute the elements of a vector space?
Soln: Meaning can be given to (a) adding two sets of amplitudes and (b) multiplying a set by an
arbitrary complex number.
1.3 V ′ is the dual space of the vector space V . For a mathematician, what objects comprise V ′?
Soln: The linear functions on V .
1.4 In quantum mechanics, what objects are the members of the vector space V ? Give an example
for the case of quantum mechanics of a member of the dual space V ′ and explain how members of
V ′ enable us to predict the outcomes of experiments.
Soln: Any complete set of amplitudes is a member of V . Since a complete set of amplitudes
uniquely specifies the physical state of the system, we can consider that physical states are the
members of V . V ′ is inhabited by bras. A bra is a function that extracts from a ket the amplitude
for a specific event. For example⟨ E2| extracts from | ψ⟩ the amplitude ⟨ E2 |ψ⟩ that a measurement
of energy determines that the system is in its second excited state.
1.5 Given that |ψ⟩ = eiπ/5|a⟩ + eiπ/4|b⟩, express ⟨ψ| as a linear combination of ⟨a| and ⟨b|.
Soln:
⟨ψ| = e−iπ/5⟨a| + e−iπ/4⟨b|
1.6 What properties characterise the bra ⟨a| that is associated with the ket |a⟩?
Soln: If we choose any orthonormal basis that includes |a⟩, then ⟨a| is the linear function that
vanishes on every basis function except |a⟩.
1.7 An electron can be in one of two potential wells that are so close that it can ‘tunnel’ from one
to the other (see §5.2 for a description of quantum-mechanical tunnelling). Its state vector can be
written
|ψ⟩ = a|A⟩ + b|B⟩, (1.1)
where |A⟩ is the state of being in the first well and |B⟩ is the state of being in the second well and all
kets are correctly normalised. What is the p√robability of finding the particle in the first well given
that: (a) a = i/2; (b) b = eiπ; (c) b = 13 + i/ 2?
Soln: (a) P = 1 ; (b) P = 0; (c) P = 7 .
4 18
1.8 An electron can ‘tunnel’ between potential wells that form a chain, so its state vector can be
written
∞
Σ
|ψ⟩ = an|n⟩, (1.2a)
−∞
where |n⟩ is the state of being in the n well, where n increases from left to right. Let
th
|n|/2
1 −i
an = √ einπ . (1.2b)
2 3
a. What is the probability of finding the electron in the nth well?
b. What is the probability of finding the electron in well 0 or anywhere to the right of it?
Σ ∞
Soln: (a) P = 21 3−|n|. (b) n =1 3−n = 21 and P0 = 12 so P (n ≥ 0) = 34.
,2 Problems
Problems
2.1 How is a wavefunction ψ(x) written in Dirac’s notation? What’s the physical significance of
the complex number ψ(x) for given x?
Soln: ψ(x) ≡ ⟨x|ψ⟩. ψ(x) is the amplitude to be found at x.
⟨ Q
2.2 Let Q be an operator. Under what circumstances is the complex number a | b| ⟩equal to the
complex number (⟨ b| Q|a⟩ )∗ for any states | a⟩ and | b⟩ ?
Soln: When Q is Hermitian.
2.3 Let Q be the operator of an observable and let ψ | ⟩ be the state of our system.
a. What are the physical interpretations of ⟨ ψ| Q|ψ⟩ and |⟨qn |ψ⟩|2, where |qn ⟩is the nth eigenket
of the observable Q and qn is the corresponding eigenvalue?
Σ
b. What is the Σ operator n |qn⟩⟨qn|, where the sum is over all eigenkets of Q? What is the
operator n qn|qn⟩⟨qn|?
c. If un(x) is the wavefunction of the state |qn⟩, write down an integral that evaluates to ⟨qn|ψ⟩.
Soln: (a) ⟨ψ|Q|ψ⟩ is the e xp ecΣ tation value of Q. |⟨qn|ψ⟩|2 is the probability of∫obtaining qn on
measuring the observable Q. (b) n |qn⟩⟨qn| is the identity operator. (c) ⟨qn|ψ⟩ = dx u∗n (x)ψ(x).
2.4 What does it mean to say that two operators commute? What is the significance of two
observables having mutually commuting operators?
Given that the commutator [P, Q] = 0 for some observables P and Q, does it follow that for all
|ψ⟩ /= 0 we have [P, Q]| ψ⟩ /= 0?
Soln: Commutation implies you always get the same result no matter in which order you use the
operators. If observables have commuting operators, there is a complete set of states in which both
observables have well-defined values.
No, [P, Q] /= 0 implies only that there is some state |φ⟩ such that PQ|φ⟩ /= QP |φ⟩; one may
readily construct an example of operators such that PQ|i⟩ = QP |i⟩ for basis vectors |i⟩ with i = 2, ∞
but PQ|0⟩ = / QP |0⟩ and PQ|1⟩ /= QP |1⟩.
2.5 Let ψ(x, t) be the correctly normalised wavefunction of a particle of mass m and potential
energy V (x). Write down expressions for the expectation values of (a) x; (b) x2; (c) the momentum
px; (d) p2x; (e) the energy.
What is the probability that the particle will be found in the interval (x1, x2)?
∫∞ ∫∞
Soln: (a) ⟨x⟩ = −∞ dxx|ψ|2; (b) ⟨x2⟩ = −∞ dxx2|ψ|2;
∫∞ ∫ ∞
∫∞
dψ dxψ∗ −h̄ d ψ + V (x)ψ .
2 2 2
(c) ⟨p⟩ = − i h̄ dxψ ∗ ; (d) ⟨p2 ⟩ = −h̄ 2 dxψ ∗ d ψ ; (e) ⟨E⟩ =
dx −∞ dx2 −∞ 2m dx2
∫ x2−∞
P (x 1, x 2) = x dx |ψ| .2
1
2.6 Write down the time-independent (TISE) and the time-dependent (TDSE) Schrödinger equa-
tions. Is it necessary for the wavefunction of a system to satisfy the TDSE? Under what circumstances
does the wavefunction of a system satisfy the TISE?
Soln:
∂|ψ⟩
TISE : H|En⟩ = En|En⟩; TDSE : i h̄ = H|ψ⟩.
∂t
The state of every system must satisfy the TDSE, whereas the TISE is satisfied only if the state has
well defined energy (which for real systems is never the case).
2.7 Why is the TDSE first-order in time, rather than second-order like Newton’s equations of
motion?
Soln: Because ψ| is ⟩ a complete set of information, so for physics to be possible it must contain
complete initial conditions for solution of the time-evolution equation. If the TDSE were second-order
in t, this condition would be violated.
2.8 A particle is confined in a potential well such that its allowed energies are En = n2E , where
n = 1, 2, . . . is an integer and E a positive constant. The corresponding energy eigenstates are |1⟩, |2⟩,
. . . , |n⟩, . . . At t = 0 the particle is in the state
|ψ(0)⟩ = 0.2|1⟩ + 0.3|2⟩ + 0.4|3⟩ + 0.843|4⟩. (2.1)
a. What is the probability, if the energy is measured at t = 0, of finding a number smaller than
6E ?
, Problems 3
b. What is the mean value and what is the rms deviation of the energy of the particle in the state
|ψ(0) ⟩?
c. Calculate the state vector |ψ⟩ at time t. Do the results found in (a) and (b) for time t remain
valid for arbitrary time t?
d. When the energy is measured it turns out to be 16 E. After the measurement, what is the state
of the system? What result is obtained if the energy is measured again?
Soln: (a) 0.13; (b) the probabilities are P1 = 0.04, P2 = 0.09, P3 = 0.16, P4 = 0.71 so ⟨E⟩ = 13.2E,
⟨E2⟩ = 196E2, and σE = 4.67E. (c) |ψ(t)⟩ = 0.2e−i E t/h̄ |1⟩ + 0.3e−i4E t/h̄ |2⟩ + 0.4e−i9E t/h̄ |3⟩ +
0.843e−i16E t/h̄ |4⟩. (d) Thereafter |ψ⟩ = |4⟩ and E = 16E with certainty.
2.9 A system has a time-independent Hamiltonian that has spectrum{ En} . Prove that the prob-
ability Pk that a measurement of energy will yield the value Ek is is time-independent. Hint: you
can do this either from Ehrenfest’s theorem, or by differentiating⟨ Ek, t| ψ⟩ w.r.t. t and using the
TDSE.
Soln: This follows from Ehrenfest’s theorem: the expectation value of any function of H is con-
stant, so the probability distribution defined by the Pk is time-independent. For a more direct
answer, consider the amplitude ak(t) to get Ek: ak = ⟨Ek, t|ψ⟩, so using the TDSE and its adjoint
∂ak ∂⟨Ek, t| ∂|ψ⟩ 1
= |ψ⟩ + ⟨E k, t| = (−⟨E k, t|H|ψ⟩ + ⟨E k, t|H|ψ⟩) = 0.
∂t ∂t ∂t ih̄
2.10 Let ψ(x) be a properly normalised wavefunction and Q an operator on wavefunctions. Let
{qr } be the spectrum of Q and u{r(x) be} the corresponding correctly normalised eigenfunctions.
Write down an expression for the probability that a measurement of Q will yield the value qr. Show
Σ ∫ −∞ ψ ∗ Q̂ψ dx.1
that
Soln: r P (qr|ψ) = 1. Show further that the expectation of Q is ⟨Q⟩ ≡ ∞
∫ 2
Pi = dxui∗(x)ψ(x) where Qui(x) = qiui(x).
! ,
∫ ∫ Σ Σ Σ ∫ Σ Σ
1 = dx |ψ|2 = dx ai∗ui∗ ajuj = ai∗aj dxui∗uj = |ai|2 = Pi
i j ij i i
Similarly
! ,
∫ ∫ Σ Σ Σ ∫
dxψ∗Qψ = dx ai∗ui∗ Q ajuj = ai∗aj dxui∗Quj
i j ij
Σ ∫ Σ Σ
= ai∗ajqj dxui∗uj = |ai|2qi = Piqi
ij i i
which is by definition the expectation value of Q.
2.11 Find the energy of neutron, electron and electromagnetic waves of wavelength 0.1 nm.
Soln:
p2 (h/λ)2 mn
En = = = 1.32 × 10−20 J = 0.0821 eV; E e=
m E = 1837En = 150.8 eV;
2m 2m e n
Eγ = hν = hc/λ = 1.988 × 10−15 J = 12.407 keV.
2.12 Neutrons are emitted from an atomic pile with a Maxwellian distribution of velocities for
temperature 400 K. Find the most probable de Broglie wavelength in the beam.
Soln: The rate at which a neutron hits the wall of the pile is proportional to its momentum p, so the
2 2
probability density of neutrons hitting the wall is proportional to p3e−p /2mkT = (2mkT )3/2x3e−x ,
√
√ p = 2mkT x.√The probability density is extremised when (3x2 − 2x4)e−x = 0, √
2
where i.e., when
x = 3/2 and p = 3mkT . Thus the most probable de Broglie wavelength is λ = h/ 3mkT =
0.126 nm.
1 In an elegant formulation of quantum mechanics, this last result is the basic postulate of the theory, and one
derives other rules for the physical interpretation of the qn, an, etc., from it – see J. von Neumann, Mathematical
Foundations of Quantum Mechanics, Princeton University Press.
