Chapter 2, Operators, measurement and time evolution Video Solutions, The Physics of Quantum Mechanics

Problem 1

How is a wavefunction $\psi(x)$ written in Dirac's notation? What's the physical significance of the complex number $\psi(x)$ for given $x$ ?

Dominador Tan

Numerade Educator

01:51

Problem 2

Mahipal Kumawat

Numerade Educator

08:02

Let $Q$ be the operator of an observable and let $|\psi\rangle$ be the state of our system.
a. What are the physical interpretations of $(\psi|Q| \psi)$ and $\left|\left\langle q_{n} \mid \psi\right\rangle\right|^{2}$, where $\left|q_{n}\right\rangle$ is the $n^{\text {th }}$ eigenket of the observable $Q$ and $q_{n}$ is the corresponding. eigenvalue?
b. What is the operator $\sum_{n}\left|q_{n}\right\rangle\left\langle q_{n}\right|$, where the sum is over all eigenkets of $Q ?$ What is the operator $\sum_{n} q_{n}\left|q_{n}\right\rangle\left\langle q_{n}\right| ?$
c. If $u_{n}(x)$ is the wavefunction of the state $\left|q_{n}\right\rangle$, write down an integral that evaluates to $\left(q_{n}|\psi\rangle\right.$.

Hafiz Shahzaib

Numerade Educator

01:42

Problem 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 $[P, Q] \neq 0$ for some observables $P$ and $Q$, does it follow that for all $|\psi\rangle \neq 0$ we have $[P, Q[|\psi\rangle \neq 0 ?$

Dominador Tan

Numerade Educator

02:47

Problem 5

Let $\psi(x, t)$ 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) $x^{2} ;$ (c) the momentum $p_{x} ;$ (d) $p_{x}^{2} ;$ (e) the energy.
What is the probability that the particle will be found in the interval $\left(x_{1}, x_{2}\right) ?$

Robert Zaballa

Numerade Educator

03:46

Problem 6

Write down the time-independent (TISE) and the time-dependent (TDSE) Schrödinger equations. 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?

Sheh Lit Chang

University of Washington

01:41

Problem 7

Why is the TDSE first-order in time, rather than second-order like Newton's equations of motion?

Surendra Kumar

Numerade Educator

09:21

Problem 8

A particle is confined in a potential well such that its allowed energies are $E_{n}=n^{2} \mathcal{E}$, where $n=1,2, \ldots$ is an integer and $\mathcal{E}$ a positive constant. The corresponding energy eigenstates are $|1\rangle,|2\rangle, \ldots,|n\rangle, \ldots$ At $t=0$ the particle is in the state
$$
|\psi(0)\rangle=0.2|1\rangle+0.3|2\rangle+0.4|3\rangle+0.843|4\rangle
$$
a. What is the probability, if the energy is measured at $t=0$, of finding a number smaller than $6 \mathcal{E} ?$
b. What is the mean value and what is the rms deviation of the energy of the particle in the state $|\psi(0)\rangle ?$
c. Calculate the state vector $|\psi\rangle$ 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 $16 \mathcal{E}$. After the measurement, what is the state of the system? What result is obtained if the energy is measured again?

Ren Jie Tuieng

Numerade Educator

02:48

Problem 9

A system has a time-independent Hamiltonian that has spectrum $\left\{E_{n}\right\} .$ Prove that the probability $P_{k}$ that a measurement of energy will yield the value $E_{k}$ is is time-independent. Hint: you can do this either from Ehrenfest's theorem, or by differentiating $\left\langle E_{k}, t \mid \psi\right\rangle$ w.r.t. $t$ and using the TDSE.

Ben Nicholson

Numerade Educator

37:13

Problem 10

Let $\psi(x)$ be a properly normalised wavefunction and $Q$ an operator on wavefunctions. Let $\left\{q_{r}\right\}$ be the spectrum of $Q$ and $\left\{u_{r}(x)\right\}$ be the corresponding correctly normalised eigenfunctions. Write down an expression for the probability that a measurement of $Q$ will yield the value $q_{r}$. Show that $\sum_{r} P\left(q_{r} \mid \psi\right)=1$. Show further that the expectation of $Q$ is $\langle Q\rangle \equiv \int_{-\infty}^{\infty} \psi^{*} \hat{Q} \psi \mathrm{d} x \cdot^{10}$

Susan Hallstrom

Numerade Educator

04:40

Problem 11

Find the energy of neutron, electron and electromagnetic waves of wavelength $0.1 \mathrm{~nm}$.

Devi Dutta Biswajeet

Numerade Educator

02:43

Problem 12

Neutrons are emitted from an atomic pile with a Maxwellian distribution of velocities for temperature $400 \mathrm{~K}$. Find the most probable de Broglie wavelength in the beam.

Prabhakar Kumar

Numerade Educator

02:22

Problem 13

A beam of neutrons with energy $E$ runs horizontally into a crystal. The crystal transmits half the neutrons and deflects the other half vertically upwards. After climbing to height $H$ these neutrons are deflected through $90^{\circ}$ onto a horizontal path parallel to the originally transmitted beam. The two horizontal beams now move a distance $L$ down the laboratory, one distance $H$ above the other. After going distance $L$, the lower beam is deflected vertically upwards and is finally deflected into the path of the upper beam such that the two beams are co-spatial as they enter the detector. Given that particles in both the lower and upper beams are in states of well-defined momentum, show that the wavenumbers $k, k^{\prime}$ of the lower and upper beams are related by
$$
k^{\prime} \simeq k\left(1-\frac{m_{\mathrm{n}} g H}{2 E}\right)
$$
In an actual experiment (R. Colella et al., Phys. Rev. Let., $\mathbf{3 4}, 1472$, 1975) $E=0.042 \mathrm{eV}$ and $L H \sim 10^{-3} \mathrm{~m}^{2}$ (the actual geometry was slightly different). Determine the phase difference between the two beams at the detector. Sketch the intensity in the detector as a function of $H$.

Penny Riley

Numerade Educator

01:58

Problem 14

A particle moves in the potential $V(\mathrm{x})$ and is known to have energy
$E_{n}$. (a) Can it have well-defined momentum for some particular $V(\mathrm{x}) ?$
(b) Can the particle simultaneously have well-defined energy and position?

Katie Mcalpine

Numerade Educator

07:06

Problem 15

The states $\{|1\rangle,|2\rangle\}$ form a complete orthonormal set of states for a two-state system. With respect to these basis states the operator $\sigma_{y}$ has matrix
$$
\sigma_{y}=\left(\begin{array}{cc}
0 & -i \\
1 & 0
\end{array}\right)
$$
${ }^{10}$ 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 $q_{n}, a_{n}$, etc., from it -see J. von Neumann, Mathematical Foundations of Quantum Mechanics, Princeton University Press.
Could $\sigma$ be an observable? What are its eigenvalues and eigenvectors in the $\{|1\rangle,|2\rangle\}$ basis? Determine the result of operating with $\sigma_{y}$ on the state
$$
|\psi\rangle=\frac{1}{\sqrt{2}}(|1\rangle-|2\rangle)
$$

Andrew Eddins

Emory University

04:15

Problem 16

A three-state system has a complete orthonormal set of states $|1\rangle,|2\rangle,|3\rangle .$ With respect to this basis the operators $H$ and $B$ have matrices
$$
H=\hbar \omega\left(\begin{array}{ccc}
1 & 0 & 0 \\
0 & -1 & 0 \\
0 & 0 & -1
\end{array}\right) \quad B=b\left(\begin{array}{lll}
1 & 0 & 0 \\
0 & 0 & 1 \\
0 & 1 & 0
\end{array}\right)
$$
where $\omega$ and $b$ are real constants.
a. Are $H$ and $B$ Hermitian?
b. Write down the eigenvalues of $H$ and find the eigenvalues of $B$. Solve for the eigenvectors of both $H$ and $B$. Explain why neither matrix uniquely specifies its eigenvectors.
c. Show that $H$ and $B$ commute. Give a basis of eigenvectors common to $H$ and $B$.

Narayan Hari

Numerade Educator

01:32

Problem 17

Given that $A$ and $B$ are Hermitian operators, show that $i[A, B]$ is a. Hermitian onerator.

James Kiss

Numerade Educator

04:00

Problem 18

Given a ordinary function $f(x)$ and an operator $R$, the operator $f(R)$ is defined to be
$$
f(R)=\sum_{i} f\left(r_{i}\right)\left|r_{i}\right\rangle\left\langle r_{i}\right|
$$
where $r_{i}$ are the eigenvalues of $R$ and $\left|r_{i}\right\rangle$ are the associated eigenkets. Show that when $f(x)=x^{2}$ this definition implies that $f(R)=R R$, that is, that operating with $f(R)$ is equivalent to applying the operator $R$ twice. What bearing does this result have in the meaning of $\mathrm{e}^{R} ?$

Lottie Adams

Numerade Educator

01:54

Problem 19

Show that if there is a complete set of mutual eigenkets of the Hermitian operators $A$ and $B$, then $[A, B]=0$. Explain the physical significance of this result.

Manik Pulyani

Numerade Educator

01:14

Problem 20

Given that for any two operators $(A B)^{\dagger}=B^{\dagger} A^{\dagger}$, show that
$$
(A B C D)^{\dagger}=D^{\dagger} C^{\dagger} B^{\dagger} A^{\dagger}
$$

Edward Downes

Numerade Educator

02:21

Problem 21

Prove for any four operators $A, B, C, D$ that
$$
[A B C, D]=A B[C, D]+A[B, D] C+[A, D] B C
$$
Explain the similarity with the rule for differentiating a product.

Ashley High

Numerade Educator

03:54

Problem 22

Show that for any three operators $A, B$ and $C$, the Jacobi identity holds:
$$
[A,[B, C]]+[B,[C, A]]+[C,[A, B]]=0
$$

Lucas Finney

Numerade Educator

02:13

Problem 23

Show that a classical harmonic oscillator satisfies the virial equation $2\langle\mathrm{KE}\rangle=\alpha\langle\mathrm{PE}\rangle$ and determine the relevant value of $\alpha$

Suzanne W.

Numerade Educator

01:27

Problem 24

Given that the wavefunction is $\psi=A \mathrm{e}^{\mathrm{i}(k z-\omega t)}+B \mathrm{e}^{-\mathrm{i}(k z+\omega t)}$, where $A$ and $B$ are constants, show that the probability current density is
$$
\mathbf{J}=v\left(|A|^{2}-|B|^{2}\right) \hat{\mathbf{z}}
$$
where $v=\hbar k / m .$ Interpret the result physically.

Suzanne W.

Numerade Educator

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