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Ch. 6 Problems

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Basic

Medium

Advanced

Basic

B-1. Calculation of the Phase Factor for a Stationary State

Consider a two-state system whose Hamiltonian is given by

\[H = \begin{pmatrix} E_0 & -A \\ -A & E_0 \end{pmatrix}\]

and suppose the eigenstate \(|II\rangle = \frac{1}{\sqrt{2}}(|1\rangle + |2\rangle)\) has energy \(E_{II} = E_0 - A\). For the amplitude of the stationary state at time \(t\), \(C_1(t) = \frac{1}{\sqrt{2}}e^{-iE_{II}t/\hbar}\), find the value of \(C_1(t)\) at \(t = \pi\hbar/(E_0 - A)\).

Hint

Use \(e^{-i\pi} = -1\). Substitute the value of \(t\) into the exponent and simplify.

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B-2. Energy Calculation of Tunnel Splitting

The inversion frequency of an ammonia molecule is \(f = 24{,}000\;\text{MHz}\), and the energy difference satisfies \(2A = hf\). Find \(A\) in units of eV. Use \(h = 6.626 \times 10^{-34}\;\text{J·s}\) and \(1\;\text{eV} = 1.602 \times 10^{-19}\;\text{J}\).

Hint

Calculate \(A = hf/2\) and convert from J to eV. Note that \(f = 2.4 \times 10^{10}\;\text{Hz}\).

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B-3. Expansion of the Determinant in the Eigenvalue Equation

For the general 2-state Hamiltonian

\[H = \begin{pmatrix} \alpha & \beta \\ \beta^* & \gamma \end{pmatrix}\]

(where \(\alpha, \gamma\) are real numbers and \(\beta\) is a complex number), expand the eigenvalue equation \(\det(H - E\,I) = 0\) and derive the quadratic equation in \(E\).

Hint

Compute \(\det\begin{pmatrix} \alpha - E & \beta \\ \beta^* & \gamma - E \end{pmatrix} = (\alpha - E)(\gamma - E) - \beta\beta^*\). Note that \(\beta\beta^* = |\beta|^2\).

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B-4. Verification of Orthogonality of Eigenvectors

For the eigenvectors given in the text

\[|I\rangle = \frac{1}{\sqrt{2}}(|1\rangle - |2\rangle), \quad |II\rangle = \frac{1}{\sqrt{2}}(|1\rangle + |2\rangle)\]

compute \(\langle I|I\rangle\), \(\langle II|II\rangle\), and \(\langle I|II\rangle\) respectively, and verify that these form an orthonormal system. Assume \(\langle 1|1\rangle = \langle 2|2\rangle = 1\) and \(\langle 1|2\rangle = \langle 2|1\rangle = 0\).

Hint

Expand each bra-ket and use the orthonormality of the basis to compute. For example, \(\langle I|I\rangle = \frac{1}{2}(\langle 1| - \langle 2|)(|1\rangle - |2\rangle)\).

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B-5. Calculating the Time Dependence of Probability

Given that \(|\psi(0)\rangle = |1\rangle\) at \(t = 0\), use equation (6.18) from the text to calculate \(C_2(t) = \langle 2|\psi(t)\rangle\) at time \(t\), and find the probability \(P_2(t) = |C_2(t)|^2\) of being in state \(|2\rangle\). Express the result in terms of \(A\), \(\hbar\), and \(E_0\).

Hint

Use \(\langle 2|I\rangle = -1/\sqrt{2}\) and \(\langle 2|II\rangle = 1/\sqrt{2}\). The difference between the two phase factors produces oscillations. The formula \(|e^{i\theta_1} - e^{i\theta_2}|^2 = 2 - 2\cos(\theta_1 - \theta_2)\) is useful.

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B-6. Solution of \(i\hbar\,dC/dt = EC\)

Solve the differential equation \(i\hbar\,\dfrac{dC}{dt} = E\,C\) (where \(E\) is a real constant), and find \(C(t)\) subject to the condition \(C(0) = C_0\). Also show that \(|C(t)|^2\) is independent of time.

Hint

Use separation of variables. Integrate \(dC/C = -iE/(\hbar)\,dt\).

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B-7. Inverse of Basis Transformation

Using Eq. (6.17a) \(|1\rangle = \frac{1}{\sqrt{2}}(|I\rangle + |II\rangle)\) and Eq. (6.17b) \(|2\rangle = \frac{1}{\sqrt{2}}(-|I\rangle + |II\rangle)\), conversely express \(|I\rangle\) and \(|II\rangle\) in terms of \(|1\rangle\) and \(|2\rangle\), and verify that Eqs. (6.15a) and (6.15b) are reproduced.

Hint

Solve for \(|I\rangle\) and \(|II\rangle\) by treating Eqs. (6.17a) and (6.17b) as simultaneous equations. Alternatively, compute (6.17a) \(-\) (6.17b) and (6.17a) \(+\) (6.17b).

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B-8. Verification of Hermiticity

Verify that the matrix

\[M = \begin{pmatrix} 3 & 2 - i \\ 2 + i & 5 \end{pmatrix}\]

is a Hermitian matrix. That is, show that \(M_{ij}^* = M_{ji}\) holds for all \(i, j\).

Hint

Take the complex conjugate of \(M_{12} = 2 - i\) and compare it with \(M_{21} = 2 + i\). Also verify that the diagonal elements are real.

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Medium

M-1. Derivation of Hermiticity from Probability Conservation

In the time evolution equations for a two-state system

\[i\hbar\frac{dC_1}{dt} = H_{11}C_1 + H_{12}C_2, \quad i\hbar\frac{dC_2}{dt} = H_{21}C_1 + H_{22}C_2\]

calculate the time derivative \(dP/dt\) of the total probability \(P = |C_1|^2 + |C_2|^2\). Derive the conditions for \(dP/dt = 0\) to hold for arbitrary \(C_1, C_2\), namely that \(H_{11}, H_{22}\) are real and that \(H_{12}^* = H_{21}\).

Hint

Use \(\frac{d}{dt}|C_1|^2 = C_1^*\frac{dC_1}{dt} + C_1\frac{dC_1^*}{dt}\), and substitute \(dC_1/dt\) and \(dC_1^*/dt\) from the equations. Do the same for \(C_2\), and read off the conditions for \(dP/dt = 0\) from the arbitrariness of \(C_1, C_2\).

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M-2. Derivation of Rabi Oscillations

Suppose that at \(t = 0\), \(|\psi(0)\rangle = |1\rangle\) (nitrogen is "up"). Under the Hamiltonian (6.7), derive the probabilities at time \(t\):

\[P_1(t) = |\langle 1|\psi(t)\rangle|^2, \quad P_2(t) = |\langle 2|\psi(t)\rangle|^2\]

and show the following results:

\[P_1(t) = \cos^2\!\left(\frac{At}{\hbar}\right), \quad P_2(t) = \sin^2\!\left(\frac{At}{\hbar}\right)\]

Also, verify that \(P_1(t) + P_2(t) = 1\) holds at all times, and express the period \(T\) over which the system oscillates completely between states \(|1\rangle\) and \(|2\rangle\) in terms of \(A\) and \(\hbar\).

Hint

Compute \(C_1(t) = \langle 1|\psi(t)\rangle\) from equation (6.18). Using \(\langle 1|I\rangle = 1/\sqrt{2}\) and \(\langle 1|II\rangle = 1/\sqrt{2}\), factor out the common factor \(e^{-iE_0 t/\hbar}\), and the remainder takes the form \(\cos(At/\hbar)\).

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M-3. Energy Levels of an Ammonia Molecule in an Electric Field

When a uniform electrostatic field \(\mathcal{E}\) is applied, an energy contribution from the electric dipole moment \(\pm \mu\mathcal{E}\) is added depending on the position of the nitrogen atom. When the Hamiltonian becomes

\[H = \begin{pmatrix} E_0 + \mu\mathcal{E} & -A \\ -A & E_0 - \mu\mathcal{E} \end{pmatrix}\]

find the eigenvalues \(E_{\pm}\) and express them in the following form:

\[E_{\pm} = E_0 \pm \sqrt{A^2 + (\mu\mathcal{E})^2}\]

Furthermore, discuss how the energy levels behave in the limit \(\mu\mathcal{E} \ll A\) and in the limit \(\mu\mathcal{E} \gg A\).

Hint

Expand the eigenvalue equation \(\det(H - EI) = 0\). Solve \((E_0 + \mu\mathcal{E} - E)(E_0 - \mu\mathcal{E} - E) - A^2 = 0\) for \(E\). In the limiting cases, Taylor expand \(\sqrt{A^2 + x^2}\) for \(x \ll A\) or \(x \gg A\).

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M-4. Diagonalization of the Hamiltonian and Change of Matrix Representation

Using the unitary matrix

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

diagonalize the Hamiltonian (6.7) of the ammonia molecule. That is, compute \(U^\dagger H\, U\) and show that the result is

\[U^\dagger H\, U = \begin{pmatrix} E_0 + A & 0 \\ 0 & E_0 - A \end{pmatrix}\]

Also, describe the relationship between the column vectors of \(U\) and the eigenvectors \(|I\rangle\), \(|II\rangle\).

Hint

Compute \(U^\dagger = U^T\) (since \(U\) is a real matrix), then find \(U^\dagger H\), and multiply by \(U\) from the right. Verify that the first column of \(U\) corresponds to the components of \(|I\rangle\), and the second column corresponds to the components of \(|II\rangle\).

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Advanced

A-1. Oscillating Electric Field as a Time-Dependent Perturbation and Transition Probability

An oscillating electric field \(\mathcal{E}(t) = \mathcal{E}_0 \cos\omega t\) with angular frequency \(\omega\) is applied to an ammonia molecule. When described in the energy basis \(\{|I\rangle, |II\rangle\}\), the Hamiltonian can be written as

\[H = \begin{pmatrix} E_I & \mu\mathcal{E}_0\cos\omega t \\ \mu\mathcal{E}_0\cos\omega t & E_{II} \end{pmatrix}\]

where \(E_I = E_0 + A\), \(E_{II} = E_0 - A\).

Given the initial state \(|\psi(0)\rangle = |II\rangle\) (the lower energy state), find the transition probability to state \(|I\rangle\) by following the steps below.

(a) Setting \(C_I(t) = b_I(t)\,e^{-iE_I t/\hbar}\), \(C_{II}(t) = b_{II}(t)\,e^{-iE_{II}t/\hbar}\), derive the differential equations for \(b_I(t)\) and \(b_{II}(t)\).

(b) Rotating wave approximation (RWA): Near the resonance condition \(\omega \approx (E_I - E_{II})/\hbar = 2A/\hbar\), simplify the equations by neglecting the rapidly oscillating terms.

(c) Under the resonance condition \(\omega = 2A/\hbar\), solve for \(b_I(t)\) and show that the transition probability \(P_{II \to I}(t) = |b_I(t)|^2\) is given by

\[P_{II \to I}(t) = \sin^2\!\left(\frac{\mu\mathcal{E}_0\,t}{2\hbar}\right)\]

Explain why this forms the basis for stimulated emission in the ammonia maser.

Hint

(a) Substituting into Eq. (6.3) and canceling the phase factors, the equations for \(b_I\) and \(b_{II}\) contain products of \(e^{\pm i(E_I - E_{II})t/\hbar}\) and \(\cos\omega t\). Use \(\cos\omega t = (e^{i\omega t} + e^{-i\omega t})/2\). (b) Near resonance, \(e^{i(\omega - \omega_0)t}\) varies slowly while \(e^{i(\omega + \omega_0)t}\) oscillates rapidly, so the latter is neglected (\(\omega_0 = 2A/\hbar\)). (c) Under the resonance condition, the equations become constant-coefficient, and can be solved using the same technique as in D6.

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A-2. Extension to a 3-State System: Generalized Quantum Oscillations

We extend the 2-state system and consider a system where three equivalent states \(|1\rangle\), \(|2\rangle\), \(|3\rangle\) are coupled to each other with the same tunneling amplitude \(-A\). The Hamiltonian is given by

\[H = \begin{pmatrix} E_0 & -A & -A \\ -A & E_0 & -A \\ -A & -A & E_0 \end{pmatrix}\]

(a) Find all eigenvalues of this Hamiltonian. (Hint: Rewrite the matrix in the form \(E_0\,I + (-A)(J - I)\), where \(J\) is the \(3\times 3\) matrix with all entries equal to 1 and \(I\) is the identity matrix.)

(b) Find the eigenvectors corresponding to each eigenvalue and normalize them. If there is degeneracy, state its degree.

(c) If the system is in state \(|1\rangle\) at \(t = 0\), find the probability \(P_1(t)\) of finding the system in state \(|1\rangle\) at time \(t\). Discuss how the characteristics of the oscillation (frequency, completeness of amplitude) differ compared to Rabi oscillations in the 2-state system.

Hint

(a) The eigenvalues of \(J\) are \(3\) (eigenvector \((1,1,1)^T/\sqrt{3}\)) and \(0\) (2-fold degenerate, any vector orthogonal to \((1,1,1)^T\)). Rewriting as \(H = (E_0 + A)I - A\,J\), the eigenvalues of \(H\) can be obtained directly from the eigenvalues of \(J\). (b) A basis for the degenerate eigenspace can be chosen as, for example, \((1,-1,0)^T/\sqrt{2}\) and \((1,1,-2)^T/\sqrt{6}\). (c) Expand \(|1\rangle\) in energy eigenstates, attach the time evolution factor to each eigenstate, and then compute \(|\langle 1|\psi(t)\rangle|^2\).

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