Suppose a $2 \times 2$ matrix $X$ (not necessarily Hermitian or unitary) is written as

\[X=a_{0}+\sigma \cdot \mathrm{a}\]

where $a_{0}$ 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 $X_{i j}$.

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Show that the determinant of a $2 \times 2$ matrix $\sigma \cdot$ a is invariant under

\[\sigma \cdot \mathbf{a} \rightarrow \sigma \cdot \mathbf{a}^{\prime}=\exp \left(\frac{i \sigma \cdot \hat{\mathbf{n}} \phi}{2}\right) \sigma \cdot \operatorname{aexp}\left(\frac{-i \sigma \cdot \hat{\mathbf{n}} \phi}{2}\right)\]

Find $a_{k}^{\prime}$ in terms of $a_{k}$ when $\hat{n}$ is in the positive $z$ -direction, and interpret your result.

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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) $(X Y)^{\dagger}=Y^{\prime} X^{\dagger},$ where $X$ and $Y$ are operators.

(c) $\exp [i f(A)]=?$ in ket-bra form, where $A$ is a Hermitian operator 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 $\psi_{a^{\prime}}\left(\mathbf{x}^{\prime}\right)=\left\langle\mathbf{x}^{\prime} | a^{\prime}\right\rangle$.

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(a) Consider two kets $|\alpha\rangle$ and $|\beta\rangle .$ Suppose $\left\langle a^{\prime} | \alpha\right\rangle,\left\langle a^{\prime \prime} | \alpha\right\rangle, \ldots$ and $\left\langle a^{\prime} | \beta\right\rangle$

$\left\langle a^{\prime \prime} | \beta\right\rangle, \ldots$ 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 $| \alpha$ ) and $|\beta\rangle$ be $| s_{z}=h / 2$ ) and $\left|s_{x}=h / 2\right\rangle,$ respectively. Write down explicitly the square matrix that corresponds to $|\alpha\rangle\langle\beta|$ in the usual $(s_{z}$ diagonal) basis..

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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.

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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) Explain 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.

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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

\[\begin{array}{l}\left.S_{x}=\frac{\hbar}{2}(|+)\langle-|+|-\rangle\langle+|\right), \quad S_{y}=\frac{i \hbar}{2}(-|+\rangle\langle-|+|-\rangle\langle+|) \\S_{z}=\frac{\hbar}{2}(|+\rangle\langle+|-|-\rangle\langle-|)\end{array}\]

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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 $\hat{n}$ is characterized by the angles shown in the accompanying 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.]

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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 ).

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A two-state system is characterized by the Hamiltonian

\[H=H_{11}|1\rangle\left\langle 1\left|+H_{22}\right| 2\right\rangle\left\langle 2\left|+H_{12}[|1\rangle\langle 2|+| 2\rangle\langle 1|]\right.\right.\]

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 cigenkets and the corresponding energy eigenvalues. Make sure that your answer makes good sense for $H_{12}=0 .$ (You need not solve this problem from scratch. The following fact may be used without proof:

\[(\mathbf{S} \cdot \hat{\mathbf{n}})|\hat{\mathbf{n}} ;+\rangle=\frac{\hbar}{2}|\hat{\mathbf{n}} ;+\rangle\]

with $|\hat{\mathbf{n}} ;+\rangle$ given by

\[|\hat{\mathbf{n}} ;+\rangle=\cos \frac{\beta}{2}|+\rangle+e^{i \alpha} \sin \frac{\beta}{2}|-\rangle\]

where $\beta$ and $\alpha$ are the polar and azimuthal angles, respectively, that characterize $\hat{\mathbf{h}}$ The angles are defined in the figure following Problem $1.9 .$ )

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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 $\gamma=0$ $\pi / 2, \text { and } \pi .)$

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A beam of spin $\frac{1}{2}$ atoms goes through a series of Stern-Gerlach-type measurements

as follows:

(a) The first measurement accepts $s_{z}=h / 2$ atoms and rejects $s_{z}=-h / 2$ 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?

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A certain observable in quantum mechanics has a $3 \times 3$ matrix representation as follows:

\[\frac{1}{\sqrt{2}}\left(\begin{array}{ccc}0 & 1 & 0 \\1 & 0 & 1 \\0 & 1 & 0\end{array}\right)\]

(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.

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Let $A$ and $B$ be observables. Suppose the simultancous 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.

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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.

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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 $A_{1} \rightarrow L_{2}, A_{2} \rightarrow L_{x}$.

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(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

\[\left\langle x^{\prime} | \alpha\right\rangle=\left(2 \pi d^{2}\right)^{-1 / 4} \exp \left[\frac{i\langle p\rangle x^{\prime}}{\hbar}\frac{\left(x^{\prime}-\langle x\rangle\right)^{2}}{4 d^{2}}\right]\]

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).

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(a) Compute

\[\left\langle\left(\Delta S_{x}\right)^{2}\right\rangle=\left\langle S_{2}^{x}\right\rangle-\left\langle S_{x}\right\rangle^{2}\]

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.

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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.

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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=\left\{\begin{array}{ll}0 & \text { for } 0< x< a \\\infty & \text { otherwise }

\end{array}\right.\]

Do this for both the ground and excited states.

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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 that 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.

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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 that are simultancous 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?

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(a) Prove that $(1 / \sqrt{2})\left(1+i \sigma_{x}\right)$ 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.

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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 $(\left\{\left|b^{\prime}\right\rangle\right\}$ 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 1.24 ) or $x$ and $p_{x}$.

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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

\[U=\sum_{r}\left|b^{(r)}\right\rangle\left\langle a^{(r)}\right|\]

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(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.

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(a) Let $x$ and $p_{x}$ be the coordinate momentum and the 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?

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(a) On page $247,$ 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 $\left[x^{2}, p^{2}\right]_{\text {classical }}$.

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The translation operator for a finite (spatial) displacement is given by

\[g(1)=\exp \left(\frac{-i \mathbf{p} \cdot 1}{\hbar}\right)\]

where $\mathbf{p}$ is the momentum operator.

(a) Evaluate

\[\left[x_{i}, g(1)\right]\]

(b) Using (a) (or otherwise), demonstrate how the expectation value (x) changes under translation.

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In the main text we discussed the effect of $g\left(d \mathbf{x}^{\prime}\right)$ 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 $(1.6 .25)$ $(1.6 .45),$ and $|\alpha\rangle \rightarrow g\left(d \mathbf{x}^{\prime}\right)|\alpha\rangle$ 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 in finitesimal translation.

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(a) Verify $(1.7 .39 \mathrm{a})$ and $(1.7 .39 \mathrm{b})$ for the expectation value of $p$ and $p^{2}$ from the Gaussian wave packet $(1.7 .35)$

(b) Evaluate the expectation value of $p$ and $p^{2}$ using the momentum-space wave function $(1.7 .42)$.

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(a) Prove the following:

i. $\left\langle p^{\prime}|x| \alpha\right\rangle=i \hbar \frac{\partial}{\partial p^{\prime}}\left\langle p^{\prime} | \alpha\right\rangle$

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 $\phi_{\alpha}\left(p^{\prime}\right)=\left\langle p^{\prime} | \alpha\right\rangle$ and $\phi_{\beta}\left(p^{\prime}\right)=\left\langle p^{\prime} | \beta\right\rangle$ are momentum-space wave functions.

(b) What is the physical significance of

\[\operatorname{cxp}\left(\frac{i x \Xi}{\hbar}\right)\]

where $x$ is the position operator and $\Xi$ is some number with the dimension of momentum? Justify your answer.

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