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Modern Quantum Mechanics

J. J. Sakurai, Jim Napolitano

Chapter 1

Fundamental Concepts - all with Video Answers

Educators

Chapter Questions

02:38

Problem 1

A beam of silver atoms is created by heating a vapor in an oven to $1000^{\circ} \mathrm{C}$, and selecting atoms with a velocity close to the mean of the thermal distribution. The beam moves through a one-meter long magnetic field with a vertical gradient $10 \mathrm{~T} / \mathrm{m}$, and impinges a screen one meter downstream of the end of the magnet. Assuming the silver atom has spin $\frac{1}{2}$ with a magnetic moment of one Bohr magneton, find the separation distance in millimeters of the two states on the screen.

Anand Jangid
Anand Jangid
Numerade Educator
01:08

Problem 2

Prove
$$
[A B, C D]=-A C\{D, B\}+A\{C, B\} D-C\{D, A\} B+\{C, A\} D B
$$

Yujie Wang
Yujie Wang
College of San Mateo
02:47

Problem 3

For the spin $\frac{1}{2}$ state $\left|S_{x} ;+\right\rangle$, evaluate both sides of the inequality (1.146), that is
$$
\left\langle(\Delta A)^{2}\right\rangle\left\langle(\Delta B)^{2}\right\rangle \geq \frac{1}{4}|\langle[A, B]\rangle|^{2}
$$
for the operators $A=S_{x}$ and $B=S_{y}$, and show that the inequality is satisfied. Repeat for the operators $A=S_{z}$ and $B=S_{y}$.

Maxime Rossetti
Maxime Rossetti
Numerade Educator
05:35

Problem 4

Suppose a $2 \times 2$ matrix $X$ (not necessarily Hermitian, nor unitary) is written as
$$
X=a_{0}+\boldsymbol{\sigma} \cdot \mathbf{a}
$$
where the matrices $\sigma$ are given in $(3.50)$ and $a_{0}$ and $a_{1,2,3}$ are numbers.
a. How are $a_{0}$ and $a_{k}(k=1,2,3)$ related to $\operatorname{tr}(X)$ and $\operatorname{tr}\left(\sigma_{k} X\right)$ ?
b. Obtain $a_{0}$ and $a_{k}$ in terms of the matrix elements $X_{i j}$.

Jacob Fry
Jacob Fry
Numerade Educator
02:44

Problem 5

Show that the determinant of a $2 \times 2$ matrix $\sigma \cdot \mathbf{a}$ is invariant under
$$
\sigma \cdot \mathbf{a} \rightarrow \sigma \cdot \mathbf{a}^{\prime} \equiv \exp \left(\frac{i \sigma \cdot \hat{\mathbf{n}} \phi}{2}\right) \sigma \cdot \mathbf{a} \exp \left(\frac{-i \sigma \cdot \hat{\mathbf{n}} \phi}{2}\right),
$$
where the matrices $\sigma$ are given in (3.50). Find $a_{k}^{\prime}$ in terms of $a_{k}$ when $\hat{\mathbf{n}}$ is in the positive $z$-direction and interpret your result.

Hannah Wilds
Hannah Wilds
Numerade Educator
04:48

Problem 6

Using the rules of bra-ket algebra, prove or evaluate the following:
a. $\operatorname{tr}(X Y)=\operatorname{tr}(Y X)$, where $X$ and $Y$ are operators;
b. $(X Y)^{\dagger}=Y^{\dagger} X^{\dagger}$, where $X$ and $Y$ are operators;
c. $\exp [i f(A)]=?$ in ket-brã form, where $A$ is a Hermitian operátor whose eigenvalues are known;
d. $\sum_{a^{\prime}} \psi_{a^{\prime}}^{*}\left(\mathbf{x}^{\prime}\right) \psi_{a^{\prime}}\left(\mathbf{x}^{\prime \prime}\right)$, where $\psi_{a^{\prime}}\left(\mathbf{x}^{\prime}\right)=\left\langle\mathbf{x}^{\prime} \mid a^{\prime}\right\rangle .$

Diogo Caetano
Diogo Caetano
Numerade Educator
20:31

Problem 7

a. Consider two kets $|\alpha\rangle$ and $|\beta\rangle .$ Suppose $\left\langle a^{\prime} \mid \alpha\right\rangle,\left\langle a^{\prime \prime} \mid \alpha\right\rangle, \ldots$ and $\left\langle a^{\prime} \mid \beta\right\rangle,\left\langle a^{\prime \prime} \mid \beta\right\rangle, \ldots$ are all known, where $\left|a^{\prime}\right\rangle,\left|a^{\prime \prime}\right\rangle, \ldots$ form a complete set of base kets. Find the matrix representation of the operator $|\alpha\rangle\langle\beta|$ in that basis.
b. We now consider a spin $\frac{1}{2}$ system and let $|\alpha\rangle$ and $|\beta\rangle$ be $\left|S_{z} ;+\right\rangle$ and $\left|S_{x} ;+\right\rangle$, respectively. Write down explicitly the square matrix that corresponds to $|\alpha\rangle\langle\beta|$ in the usual ( $s_{z}$ diagonal) basis.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
04:31

Problem 8

Suppose $|i\rangle$ and $|j\rangle$ are eigenkets of some Hermitian operator $A$. Under what condition can we conclude that $|i\rangle+|j\rangle$ is also an eigenket of $A$ ? Justify your answer.

Melissa Munoz
Melissa Munoz
Numerade Educator
10:59

Problem 9

Consider a ket space spanned by the eigenkets $\left\{\left|a^{\prime}\right\rangle\right\}$ of a Hermitian operator $A$. There is no degeneracy.
a. Prove that
$$
\prod_{a^{\prime}}\left(A-a^{\prime}\right)
$$
is the null operator.
b. What is the significance of
$$
\prod_{a^{\prime \prime} \neq a^{\prime}} \frac{\left(A-a^{\prime \prime}\right)}{\left(a^{\prime}-a^{\prime \prime}\right)} ?
$$
c. Illustrate (a) and (b) using $A$ set equal to $S_{z}$ of a spin $\frac{1}{2}$ system.

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
01:23

Problem 10

Using the orthonormality of $|+\rangle$ and $|-\rangle$, prove
$$
\left[S_{i}, S_{j}\right]=i \varepsilon_{i j k} \hbar S_{k}, \quad\left\{S_{i}, S_{j}\right\}=\left(\frac{\hbar^{2}}{2}\right) \delta_{i j},
$$
where
$$
\begin{aligned}
S_{x} &=\frac{\hbar}{2}(|+\rangle\langle-|+|-\rangle\langle+|), \quad S_{y}=\frac{i \hbar}{2}(-|+\rangle\langle-|+|-\rangle\langle+|) \\
S_{z} &=\frac{\hbar}{2}(|+\rangle\langle+|-|-\rangle\langle-|)
\end{aligned}
$$

Carson Merrill
Carson Merrill
Numerade Educator
View

Problem 11

Construct $|\mathbf{S} \cdot \hat{\mathbf{n}} ;+\rangle$ such that
$$
\mathbf{S} \cdot \hat{\mathbf{n}}|\mathbf{S} \cdot \hat{\mathbf{n}} ;+\rangle=\left(\frac{\hbar}{2}\right)|\mathbf{S} \cdot \hat{\mathbf{n}} ;+\rangle
$$
where $\hat{\mathbf{n}}$ is characterized by the angles shown in the figure. Express your answer as a linear combination of $|+\rangle$ and $|-\rangle$. [Note: The answer is
$$
\cos \left(\frac{\beta}{2}\right)|+\rangle+\sin \left(\frac{\beta}{2}\right) e^{i \alpha}|-\rangle
$$
But do not just verify that this answer satisfies the above eigenvalue equation. Rather, treat the problem as a straightforward eigenvalue problem. Also do not use rotation operators, which we will introduce later in this book.]

Victor Salazar
Victor Salazar
Numerade Educator
09:59

Problem 12

The Hamiltonian operator for a two-state system is given by
$$
H=a(|1\rangle\langle 1|-| 2\rangle\langle 2|+| 1\rangle\langle 2|+| 2\rangle\langle 1|),
$$
where $a$ is a number with the dimension of energy. Find the energy eigenvalues and the corresponding energy eigenkets (as linear combinations of $|1\rangle$ and $|2\rangle$ ).

Isaac Huidobro
Isaac Huidobro
Numerade Educator
09:59

Problem 13

A two-state system is characterized by the Hamiltonian
$$
H=H_{11}|1\rangle\left\langle 1\left|+H_{22}\right| 2\right\rangle\langle 2|+H_{12}[|1\rangle\langle 2|+| 2\rangle\langle 1|]
$$
where $H_{11}, H_{22}$, and $H_{12}$ are real numbers with the dimension of energy, and $|1\rangle$ and $|2\rangle$ are eigenkets of some observable $(\neq H)$. Find the energy eigenkets and corresponding energy eigenvalues. Make sure that your answer makes good sense for $H_{12}=0$.

Isaac Huidobro
Isaac Huidobro
Numerade Educator
01:23

Problem 14

A spin $\frac{1}{2}$ system is known to be in an eigenstate of $\mathbf{S} \cdot \hat{\mathbf{n}}$ with eigenvalue $\hbar / 2$, where $\hat{\mathbf{n}}$ is a unit vector lying in the $x z$-plane that makes an angle $\gamma$ with the positive $z$-axis.
a. Suppose $S_{x}$ is measured. What is the probability of getting $+\hbar / 2$ ?
b. Evaluate the dispersion in $S_{x}$, that is,
$$
\left\langle\left(S_{x}-\left\langle S_{x}\right\rangle\right)^{2}\right\rangle
$$
(For your own peace of mind check your answers for the special cases $\gamma=0, \pi / 2$, and $\pi$.)

Dominador Tan
Dominador Tan
Numerade Educator
09:51

Problem 15

A beam of spin $\frac{1}{2}$ atoms goes through a series of Stern-Gerlach type measurements as follows.
a. The first measurement accepts $s_{z}=\hbar / 2$ atoms and rejects $s_{z}=-\hbar / 2$ atoms.
b. The second measurement accepts $s_{n}=\hbar / 2$ atoms and rejects $s_{n}=-\hbar / 2$ atoms, where $s_{n}$ is the eigenvalue of the operator $\mathbf{S} \cdot \hat{\mathbf{n}}$, with $\hat{\mathbf{n}}$ making an angle $\beta$ in the $x z$-plane with respect to the $z$-axis.
c. The third measurement accepts $s_{z}=-\hbar / 2$ atoms and rejects $s_{z}=\hbar / 2$ atoms.
What is the intensity of the final $s_{z}=-\hbar / 2$ beam when the $s_{z}=\hbar / 2$ beam surviving the first measurement is normalized to unity? How must we orient the second measuring apparatus if we are to maximize the intensity of the final $s_{z}=-\hbar / 2$ beam?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
05:49

Problem 16

A certain observable in quantum mechanics has a $3 \times 3$ matrix representation as follows:
$$
\frac{1}{\sqrt{2}}\left(\begin{array}{lll}
0 & 1 & 0 \\
1 & 0 & 1 \\
0 & 1 & 0
\end{array}\right)
$$
a. Find the normalized eigenvectors of this observable and the corresponding eigenvalues. Is there any degeneracy?
b. Give a physical example where all this is relevant.

Anthony Ramos
Anthony Ramos
Numerade Educator
02:43

Problem 17

Let $A$ and $B$ be observables. Suppose the simultaneous eigenkets of $A$ and $B\left\{\left|a^{\prime}, b^{\prime}\right\rangle\right\}$ form a complete orthonormal set of base kets. Can we always conclude that
$$
[A, B]=0 ?
$$
If your answer is yes, prove the assertion. If your answer is no, give a counterexample.

Nick Johnson
Nick Johnson
Numerade Educator
01:24

Problem 18

Two Hermitian operators anticommute:
$$
\{A, B\}=A B+B A=0
$$
Is it possible to have a simultaneous (that is, common) eigenket of $A$ and $B$ ? Prove or illustrate your assertion.

Victor Salazar
Victor Salazar
Numerade Educator
02:15

Problem 19

Two observables $A_{1}$ and $A_{2}$, which do not involve time explicitly, are known not to commute,
$$
\left[A_{1}, A_{2}\right] \neq 0
$$
yet we also know that $A_{1}$ and $A_{2}$ both commute with the Hamiltonian:
$$
\left[A_{1}, H\right]=0, \quad\left[A_{2}, H\right]=0 .
$$
Prove that the energy eigenstates are, in general, degenerate. Are there exceptions? As an example, you may think of the central-force problem $H=\mathbf{p}^{2} / 2 m+V(r)$, with $A_{1} \rightarrow L_{z}, A_{2} \rightarrow L_{x}$.

Chai Santi
Chai Santi
Numerade Educator
01:09

Problem 20

a. The simplest way to derive the Schwarz inequality goes as follows. First, observe
$$
\left(\left\langle\alpha\left|+\lambda^{*}\langle\beta|\right) \cdot(|\alpha\rangle+\lambda|\beta\rangle) \geq 0\right.\right.
$$
for any complex number $\lambda$; then choose $\lambda$ in such a way that the preceding inequality reduces to the Schwarz inequality.
b. Show that the equality sign in the generalized uncertainty relation holds if the state in question satisfies
$$
\Delta A|\alpha\rangle=\lambda \Delta B|\alpha\rangle
$$
with $\lambda$ purely imaginary.
c. Explicit calculations using the usual rules of wave mechanics show that the wave function for a Gaussian wave packet given by
$$
\left\langle x^{\prime} \mid \alpha\right\rangle=\left(2 \pi d^{2}\right)^{-1 / 4} \exp \left[\frac{i\langle p\rangle x^{\prime}}{\hbar}-\frac{\left(x^{\prime}-\langle x\rangle\right)^{2}}{4 d^{2}}\right]
$$
satisfies the minimum uncertainty relation
$$
\sqrt{\left\langle(\Delta x)^{2}\right\rangle} \sqrt{\left\langle(\Delta p)^{2}\right\rangle}=\frac{\hbar}{2}
$$
Prove that the requirement
$$
\left\langle x^{\prime}|\Delta x| \alpha\right\rangle=(\text { imaginary number })\left\langle x^{\prime}|\Delta p| \alpha\right\rangle
$$
is indeed satisfied for such a Gaussian wave packet, in agreement with (b).

Carson Merrill
Carson Merrill
Numerade Educator
24:10

Problem 21

a. Compute
$$
\left\langle\left(\Delta S_{x}\right)^{2}\right\rangle \equiv\left\langle S_{x}^{2}\right\rangle-\left\langle S_{x}\right\rangle^{2}
$$
where the expectation value is taken for the $S_{z}+$ state. Using your result, check the generalized uncertainty relation
$$
\left\langle(\Delta A)^{2}\right\rangle\left\langle(\Delta B)^{2}\right\rangle \geq \frac{1}{4}|\langle[A, B]\rangle|^{2},
$$
with $A \rightarrow S_{x}, B \rightarrow S_{y}$.
b. Check the uncertainty relation with $A \rightarrow S_{x}, B \rightarrow S_{y}$ for the $S_{x}+$ state.

Mirza Aslam Beig
Mirza Aslam Beig
Numerade Educator
01:15

Problem 22

Find the linear combination of $|+\rangle$ and $|-\rangle$ kets that maximizes the uncertainty product
$$
\left\langle\left(\Delta S_{x}\right)^{2}\right\rangle\left\langle\left(\Delta S_{y}\right)^{2}\right\rangle .
$$
Verify explicitly that for the linear combination you found, the uncertainty relation for $S_{x}$ and $S_{y}$ is not violated.

Suzanne W.
Suzanne W.
Numerade Educator
02:01

Problem 23

Evaluate the $x-p$ uncertainty product $\left\langle(\Delta x)^{2}\right\rangle\left\langle(\Delta p)^{2}\right\rangle$ for a one-dimensional particle confined between two rigid walls
$$
V= \begin{cases}0 & \text { for } 0<x<a \\ \infty & \text { otherwise }\end{cases}
$$
Do this for both the ground and excited states.

João Gabriel Alencar Caribé
João Gabriel Alencar Caribé
Numerade Educator
07:36

Problem 24

Estimate the rough order of magnitude of the length of time that an ice pick can be balanced on its point if the only limitation is that set by the Heisenberg uncertainty principle. Assume that the point is sharp and that the point and the surface on which it rests are hard. You may make approximations which do not alter the general order of magnitude of the result. Assume reasonable values for the dimensions and weight of the ice pick. Obtain an approximate numerical result and express it in seconds.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
20:31

Problem 25

Consider a three-dimensional ket space. If a certain set of orthonormal kets, say, $|1\rangle$, $|2\rangle$, and $|3\rangle$, are used as the base kets, the operators $A$ and $B$ are represented by
$$
A \doteq\left(\begin{array}{ccc}
a & 0 & 0 \\
0 & -a & 0 \\
0 & 0 & -a
\end{array}\right), \quad B \doteq\left(\begin{array}{ccc}
b & 0 & 0 \\
0 & 0 & -i b \\
0 & i b & 0
\end{array}\right)
$$
with $a$ and $b$ both real.
a. Obviously $A$ exhibits a degenerate spectrum. Does $B$ also exhibit a degenerate spectrum?
b. Show that $A$ and $B$ commute.
c. Find a new set of orthonormal kets which are simultaneous eigenkets of both $A$ and $B$. Specify the eigenvalues of $A$ and $B$ for each of the three eigenkets. Does your specification of eigenvalues completely characterize each eigenket?

Dr. Rajveer Singh
Dr. Rajveer Singh
Numerade Educator
01:00

Problem 26

a. Prove that $(1 / \sqrt{2})\left(1+i \sigma_{x}\right)$, where the matrix $\sigma_{x}$ is given in $(3.50)$, acting on a two-component spinor can be regarded as the matrix representation of the rotation operator about the $x$-axis by angle $-\pi / 2$. (The minus sign signifies that the rotation is clockwise.)
b. Construct the matrix representation of $S_{z}$ when the eigenkets of $S_{y}$ are used as base vectors.

Raj Bala
Raj Bala
Numerade Educator
01:22

Problem 27

Some authors define an operator to be real when every member of its matrix elements $\left\langle b^{\prime}|A| b^{\prime \prime}\right\rangle$ is real in some representation $\left(\left\{\left|b^{\prime}\right\rangle\right\}\right.$ basis in this case). Is this concept representation independent, that is, do the matrix elements remain real even if some basis other than $\left\{\left|b^{\prime}\right\rangle\right\}$ is used? Check your assertion using familiar operators such as $S_{y}$ and $S_{z}$ (see Problem 1.26) or $x$ and $p_{x}$.

Varsha Aggarwal
Varsha Aggarwal
Numerade Educator
09:36

Problem 28

Construct the transformation matrix that connects the $S_{z}$ diagonal basis to the $S_{x}$ diagonal basis. Show that your result is consistent with the general relation
$$
U=\sum_{r}\left|b^{(r)}\right\rangle\left\langle a^{(r)}\right| .
$$

Chris Trentman
Chris Trentman
Numerade Educator
04:17

Problem 29

a. Suppose that $f(A)$ is a function of a Hermitian operator $A$ with the property $A\left|a^{\prime}\right\rangle=$ $a^{\prime}\left|a^{\prime}\right\rangle$. Evaluate $\left\langle b^{\prime \prime}|f(A)| b^{\prime}\right\rangle$ when the transformation matrix from the $a^{\prime}$ basis to the $b^{\prime}$ basis is known.
b. Using the continuum analogue of the result obtained in (a), evaluate
$$
\left\langle\mathbf{p}^{\prime \prime}|F(r)| \mathbf{p}^{\prime}\right\rangle
$$
Simplify your expression as far as you can. Note that $r$ is $\sqrt{x^{2}+y^{2}+z^{2}}$, where $x$, $y$, and $z$ are operators.

Victor Salazar
Victor Salazar
Numerade Educator
01:16

Problem 30

a. Let $x$ and $p_{x}$ be the coordinate and linear momentum in one dimension. Evaluate the classical Poisson bracket
$$
\left[x, F\left(p_{x}\right)\right]_{\text {classical }}
$$
b. Let $x$ and $p_{x}$ be the corresponding quantum-mechanical operators this time. Evaluate the commutator
$$
\left[x, \exp \left(\frac{i p_{x} a}{\hbar}\right)\right]
$$
c. Using the result obtained in (b), prove that
$$
\exp \left(\frac{i p_{x} a}{\hbar}\right)\left|x^{\prime}\right\rangle \quad\left(x\left|x^{\prime}\right\rangle=x^{\prime}\left|x^{\prime}\right\rangle\right)
$$
is an eigenstate of the coordinate operator $x$. What is the corresponding eigenvalue?

Lottie Adams
Lottie Adams
Numerade Educator
01:02

Problem 31

a. On p. 247, Gottfried (1966) states that
$$
\left[x_{i}, G(\mathbf{p})\right]=i \hbar \frac{\partial G}{\partial p_{i}}, \quad\left[p_{i}, F(\mathbf{x})\right]=-i \hbar \frac{\partial F}{\partial x_{i}}
$$
can be "easily derived" from the fundamental commutation relations for all functions of $F$ and $G$ that can be expressed as power series in their arguments. Verify this statement.
b. Evaluate $\left[x^{2}, p^{2}\right]$. Compare your result with the classical Poisson bracket $\left[x^{2}, p^{2}\right]_{\text {classical }} .$

Carson Merrill
Carson Merrill
Numerade Educator
01:16

Problem 32

The translation operator for a finite (spatial) displacement is given by
$$
\mathscr{J}(\mathbf{l})=\exp \left(\frac{-i \mathbf{p} \cdot \mathbf{l}}{\hbar}\right)
$$
where $\mathbf{p}$ is the momentum operator.
a. Evaluate
$$
\left[x_{i}, \mathscr{J}(\mathbf{l})\right] .
$$
b. Using (a) (or otherwise), demonstrate how the expectation value $\langle\mathbf{x}\rangle$ changes under translation.

Lottie Adams
Lottie Adams
Numerade Educator
02:44

Problem 33

In the main text we discussed the effect of $\mathscr{J}\left(d \mathbf{x}^{\prime}\right)$ on the position and momentum eigenkets and on a more general state ket $|\alpha\rangle .$ We can also study the behavior of expectation values $\langle\mathbf{x}\rangle$ and $\langle\mathbf{p}\rangle$ under infinitesimal translation. Using (1.207), (1.227), and $|\alpha\rangle \rightarrow \mathscr{J}\left(d \mathbf{x}^{\prime}\right)|\alpha\rangle$ only, prove $\langle\mathbf{x}\rangle \rightarrow\langle\mathbf{x}\rangle+d \mathbf{x}^{\prime},\langle\mathbf{p}\rangle \rightarrow\langle\mathbf{p}\rangle$ under infinitesimal translation.

Aniket Bajaj
Aniket Bajaj
Numerade Educator
01:16

Problem 34

Starting with a momentum operator $\mathbf{p}$ having eigenstates $\left|\mathbf{p}^{\prime}\right\rangle$, define an infinitesimal boost operator $\mathscr{B}\left(d \mathbf{p}^{\prime}\right)$ that changes one momentum eigenstate into another, that is
$$
\mathscr{B}\left(d \mathbf{p}^{\prime}\right)\left|\mathbf{p}^{\prime}\right\rangle=\left|\mathbf{p}^{\prime}+d \mathbf{p}^{\prime}\right\rangle .
$$
Show that the form $\mathscr{B}\left(d \mathbf{p}^{\prime}\right)=1+i \mathbf{W} \cdot d \mathbf{p}^{\prime}$, where $\mathbf{W}$ is Hermitian, satisfies the unitary, associative, and inverse properties that are appropriate for $\mathscr{B}\left(d \mathbf{p}^{\prime}\right)$. Use dimensional analysis to express $\mathbf{W}$ in terms of the position operator $\mathbf{x}$, and show that the result satisfies the canonical commutation relations $\left[x_{i}, p_{j}\right]=i \hbar \delta_{i j}$. Derive an expression for the matrix element $\left\langle\mathbf{p}^{\prime}|\mathbf{x}| \alpha\right\rangle$ in terms of a derivative with respect to $\mathbf{p}^{\prime}$ of $\left\langle\mathbf{p}^{\prime} \mid \alpha\right\rangle$.

Lottie Adams
Lottie Adams
Numerade Educator
05:21

Problem 35

a. Verify $(1.271 \mathrm{a})$ and $(1.271 \mathrm{~b})$ for the expectation value of $p$ and $p^{2}$ from the Gaussian wave packet (1.267).
b. Evaluate the expectation value of $p$ and $p^{2}$ using the momentum-space wave function (1.274).

Timothy James
Timothy James
Numerade Educator
01:45

Problem 36

a. Prove the following:
(i) $\left\langle p^{\prime}|x| \alpha\right\rangle=i \hbar \frac{\partial}{\partial p^{\prime}}\left\langle p^{\prime} \mid \alpha\right\rangle$,
(ii) $\langle\beta|x| \alpha\rangle=\int d p^{\prime} \phi_{\beta}^{*}\left(p^{\prime}\right) i \hbar \frac{\partial}{\partial p^{\prime}} \phi_{\alpha}\left(p^{\prime}\right)$,
where $\phi_{\alpha}\left(p^{\prime}\right)=\left\langle p^{\prime} \mid \alpha\right\rangle$ and $\phi_{\beta}\left(p^{\prime}\right)=\left\langle p^{\prime} \mid \beta\right\rangle$ are momentum-space wave functions.
b. What is the physical significance of
$$
\exp \left(\frac{i x \Xi}{\hbar}\right)
$$
where $x$ is the position operator and $\Xi$ is some number with the dimension of momentum? Justify your answer.

Suzanne W.
Suzanne W.
Numerade Educator