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Ch. 5 Problems

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Table of Contents

Basic

Medium

Advanced

Basic

B-1. Verifying the Normalization Condition

Verify that the state \(|\psi\rangle = \frac{1+i}{2}|+\rangle + \frac{\sqrt{2}}{2}|-\rangle\) satisfies the normalization condition \(|c_+|^2 + |c_-|^2 = 1\).

Hint

For a complex number \(z = a + bi\), we have \(|z|^2 = a^2 + b^2\). To calculate the absolute value squared of \(c_+ = \frac{1+i}{2}\), compute \(|c_+|^2 = c_+^* c_+ = \frac{1-i}{2} \cdot \frac{1+i}{2}\).

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B-2. Calculating Inner Products

For \(|\psi\rangle = \frac{1}{\sqrt{3}}|+\rangle + \sqrt{\frac{2}{3}}\,e^{i\pi/4}|-\rangle\), find the inner products \(\langle+|\psi\rangle\) and \(\langle-|\psi\rangle\), and calculate the absolute value squared of each.

Hint

Using orthonormality \(\langle+|+\rangle = 1\), \(\langle+|-\rangle = 0\), you can directly obtain \(\langle+|\psi\rangle = c_+\). Recall that \(|e^{i\theta}|^2 = 1\).

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B-3. Action of Outer Products (Projection Operators)

Apply the projection operator \(\hat{P}_+ = |+\rangle\langle+|\) to the state \(|\psi\rangle = \frac{3}{5}|+\rangle + \frac{4}{5}|-\rangle\). Write the result as a linear combination of \(|+\rangle\) and \(|-\rangle\), and verify whether the resulting state is normalized.

Hint

We have \(\hat{P}_+|\psi\rangle = |+\rangle\langle+|\psi\rangle = |+\rangle \cdot c_+\). Note that the state after projection is generally not normalized.

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B-4. Expansion in the \(x\) Basis

Expand the state \(|+\rangle\) (spin up in the \(z\) direction) in terms of the \(x\)-direction basis \(|+\rangle_x\), \(|-\rangle_x\). That is, find the coefficients \(a\) and \(b\) in

\[|+\rangle = a\,|+\rangle_x + b\,|-\rangle_x\]

using equations (5.11) and (5.12).

Hint

Treat equations (5.11) and (5.12) as a system of simultaneous equations and solve for \(|+\rangle\) and \(|-\rangle\). Alternatively, act on both sides from the left with \({}_x\langle+|\) to obtain \(a = {}_x\langle+|+\rangle\). Use the fact that \(|+\rangle_x\) and \(|-\rangle_x\) are orthonormal.

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B-5. Orthogonality of the \(y\) basis

Using equations (5.13) and (5.14), compute the inner product \({}_y\langle+|-\rangle_y\) and verify that \(|+\rangle_y\) and \(|-\rangle_y\) are orthogonal.

Hint

The bra corresponding to \(|+\rangle_y = \frac{1}{\sqrt{2}}|+\rangle + \frac{i}{\sqrt{2}}|-\rangle\) is \({}_y\langle+| = \frac{1}{\sqrt{2}}\langle+| + \left(\frac{i}{\sqrt{2}}\right)^*\langle-| = \frac{1}{\sqrt{2}}\langle+| - \frac{i}{\sqrt{2}}\langle-|\). Remember to take the complex conjugate of the coefficients when constructing the bra.

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B-6. Finding Probabilities from Probability Amplitudes

A particle with spin up in the \(y\) direction (state \(|+\rangle_y\)) is passed through a Stern-Gerlach apparatus oriented in the \(z\) direction. Find the probability of obtaining \(S_z = +\hbar/2\) and the probability of obtaining \(S_z = -\hbar/2\), respectively.

Hint

Read off the expansion coefficients of \(|+\rangle_y\) in the \(z\) basis from Eq. (5.13), and calculate the square of the absolute value of each coefficient. Consider what \(|i/\sqrt{2}|^2\) equals.

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B-7. Using the Kronecker Delta

Using the Kronecker delta \(\delta_{ij}\) defined for \(i, j \in \{+, -\}\), compute the following sum:

\[\sum_{j \in \{+,-\}} \delta_{+j}\,\delta_{j-}\]
Hint

Find the value of \(\delta_{+j}\,\delta_{j-}\) for each case \(j = +\) and \(j = -\), then add them together. Use \(\delta_{++} = 1\), \(\delta_{+-} = 0\), etc.

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B-8. Components of the Basis Transformation Matrix

Using equations (5.13) and (5.14), write out each component of the basis transformation matrix from the \(z\) basis to the \(y\) basis:

\[U = \begin{pmatrix} \langle+|+\rangle_y & \langle+|-\rangle_y \\ \langle-|+\rangle_y & \langle-|-\rangle_y \end{pmatrix}\]
Hint

From \(|+\rangle_y = \frac{1}{\sqrt{2}}|+\rangle + \frac{i}{\sqrt{2}}|-\rangle\), one can directly read off \(\langle+|+\rangle_y\) and \(\langle-|+\rangle_y\). The same applies for \(|-\rangle_y\).

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Medium

M-1. Deriving the Normalization Condition from the Completeness Relation

Using the completeness relation \(|+\rangle\langle+| + |-\rangle\langle-| = \mathbf{1}\) and the definition of the inner product, prove that for any normalized state \(|\psi\rangle\) (\(\langle\psi|\psi\rangle = 1\)),

\[|\langle+|\psi\rangle|^2 + |\langle-|\psi\rangle|^2 = 1\]

holds.

Hint

Insert the completeness relation \(\mathbf{1} = |+\rangle\langle+| + |-\rangle\langle-|\) between \(|\psi\rangle\) and \(\langle\psi|\) in the left-hand side of \(\langle\psi|\psi\rangle = 1\). That is, expand \(\langle\psi|\mathbf{1}|\psi\rangle\).

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M-2. Completeness Relation for the \(x\) Basis

Using Equations (5.11) and (5.12), verify the completeness relation for the \(x\)-direction basis

\[|+\rangle_x\,{}_x\langle+| + |-\rangle_x\,{}_x\langle-| = \mathbf{1}\]

by computing the matrix representation in the \(z\) basis.

Hint

Represent \(|+\rangle_x\) as the column vector \(\frac{1}{\sqrt{2}}\begin{pmatrix}1\\1\end{pmatrix}\) and \({}_x\langle+|\) as the row vector \(\frac{1}{\sqrt{2}}\begin{pmatrix}1 & 1\end{pmatrix}\). The outer product \(|+\rangle_x\,{}_x\langle+|\) becomes a \(2\times 2\) matrix. Compute \(|-\rangle_x\,{}_x\langle-|\) in the same way, and show that adding the two matrices gives the identity matrix \(\begin{pmatrix}1&0\\0&1\end{pmatrix}\).

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M-3. Unitarity of the Basis Transformation Matrix

Show that the basis transformation matrix from Eq. (5.15)

\[U = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix}\]

is a unitary matrix, i.e., that it satisfies \(U^\dagger U = \mathbf{1}\). Here \(U^\dagger\) is the transpose complex conjugate (Hermitian conjugate) of \(U\).

Hint

Since all elements of \(U\) are real, \(U^\dagger = U^T\) (the transpose matrix). Simply compute \(U^T U\) and verify that it equals the \(2\times 2\) identity matrix.

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M-4. Probability of Sequential Measurements

Consider the following sequential Stern-Gerlach experiment:

  1. An apparatus oriented in the \(z\) direction selects only the beam with spin up (\(S_z = +\hbar/2\)).
  2. The selected beam is passed through an apparatus oriented in the \(x\) direction.
  3. From the \(x\) direction apparatus, only the beam with \(S_x = +\hbar/2\) is selected.
  4. The selected beam is passed through a \(z\) direction apparatus again.

Find the probability of obtaining \(S_z = -\hbar/2\) at the final stage by tracing the probability amplitudes step by step through each stage.

Hint

Use the "multiply amplitudes then add" rule from Ch. 4. The state after step 1 is \(|+\rangle\). The amplitude for \(|+\rangle_x\) to be selected in step 2 is \({}_x\langle+|+\rangle\). The amplitude for \(|-\rangle\) to be found in step 4 is \(\langle-|+\rangle_x\). The overall amplitude is the product of these.

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Advanced

A-1. Spin Eigenstates in an Arbitrary Direction

Consider a unit vector \(\hat{\mathbf{n}} = (\sin\theta\cos\phi,\;\sin\theta\sin\phi,\;\cos\theta)\) pointing in the direction specified by polar angle \(\theta\) and azimuthal angle \(\phi\) from the \(z\) axis. Verify through the following steps that the eigenstate corresponding to the spin component \(S_{\hat{n}} = +\hbar/2\) along the \(\hat{\mathbf{n}}\) direction is given by

\[|+\rangle_{\hat{n}} = \cos\frac{\theta}{2}\,|+\rangle + e^{i\phi}\sin\frac{\theta}{2}\,|-\rangle\]

(a) Confirm that when \(\theta = 0\), \(|+\rangle_{\hat{n}} = |+\rangle\), and when \(\theta = \pi\), \(|+\rangle_{\hat{n}} = e^{i\phi}|-\rangle\) (which is \(|-\rangle\) up to an overall phase).

(b) Confirm that when \(\theta = \pi/2,\;\phi = 0\), the expression reproduces \(|+\rangle_x\) from Eq. (5.11).

(c) Confirm that when \(\theta = \pi/2,\;\phi = \pi/2\), the expression reproduces \(|+\rangle_y\) from Eq. (5.13).

(d) Show that this state is normalized (\({}_{\hat{n}}\langle+|+\rangle_{\hat{n}} = 1\)).

(e) Show that when this state is measured along the \(z\) direction, the probability of obtaining \(S_z = +\hbar/2\) is \(\cos^2(\theta/2)\), and discuss the geometric meaning of \(\theta\).

Hint

(a)–(c) simply require substituting the values of \(\theta\) and \(\phi\). For (d), compute \(|\cos(\theta/2)|^2 + |e^{i\phi}\sin(\theta/2)|^2\). The probability in (e) is \(|\langle+|+\rangle_{\hat{n}}|^2 = \cos^2(\theta/2)\), which corresponds to the square of the cosine of "half the angle" between \(|+\rangle\) (the north pole) and \(|+\rangle_{\hat{n}}\) on the Bloch sphere.

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A-2. "Which Path Was Taken" and the Disappearance of Interference

Consider the following thought experiment.

Setup: After separating a silver atom beam into \(|+\rangle\) and \(|-\rangle\) using a Stern-Gerlach apparatus in the \(z\) direction, the two beams are recombined without being blocked, and then measured with a Stern-Gerlach apparatus in the \(x\) direction.

(a) Consider the case where the two beams are completely recombined and it is impossible to distinguish which path was taken. When the initial state is \(|+\rangle_x\), calculate the probability of obtaining \(S_x = +\hbar/2\) in the final measurement, using the "add amplitudes first, then take the absolute value squared" rule from Ch. 4.

(b) Now suppose that only the \(|+\rangle\) path is marked so that "which path was taken can be distinguished." In this case, the "add probabilities" rule applies. Calculate the probability of obtaining \(S_x = +\hbar/2\) and compare with the result of (a).

(c) Explain the difference between the results of (a) and (b) by explicitly showing the "interference terms," and discuss how "acquiring which-path information destroys interference" from the perspective of the probability amplitude rules in Ch. 4.

Hint

(a) Expand the initial state \(|+\rangle_x\) in the \(z\) basis, and add the amplitudes going through intermediate states \(|+\rangle\), \(|-\rangle\): amplitude \(= \sum_{j=\pm} {}_x\langle+|j\rangle\langle j|+\rangle_x\). Recall the completeness relation. (b) When the paths are distinguishable, add probabilities: \(P = \sum_{j=\pm} |{}_x\langle+|j\rangle|^2\,|\langle j|+\rangle_x|^2\). (c) The difference between the two corresponds to the interference terms (cross terms).

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