Skip to content

Ch. 7 Problems

← Back to chapter　|　View solutions

Table of Contents

Basic

Medium

Advanced

Basic

B-1. Substituting a Plane Wave into the Free-Particle Schrödinger Equation

Substitute the plane wave \(\Psi(x,t) = Ae^{i(kx - \omega t)}\) into the free-particle Schrödinger equation

\[i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2}\]

and determine the dispersion relation \(\omega = \omega(k)\).

Hint

Use the fact that on the left-hand side \(\partial\Psi/\partial t = -i\omega\Psi\), and on the right-hand side \(\partial^2\Psi/\partial x^2 = -k^2\Psi\). Dividing both sides by \(\Psi\) yields the relation between \(\omega\) and \(k\).

→ View solution

B-2. Apply the momentum operator to each of the following wave functions and find the result. If it is an eigenfunction, state the eigenvalue.

Apply the momentum operator \(\hat{p} = -i\hbar\frac{\partial}{\partial x}\) to each of the following wave functions and find the result. If it is an eigenfunction, state the eigenvalue.

(a) \(\psi(x) = e^{5ix/\hbar}\)

(b) \(\psi(x) = \cos(kx)\)

(c) \(\psi(x) = (x^2 + 1)e^{ipx/\hbar}\)

Hint

(a) involves differentiation of an exponential function. (b) becomes clearer if you write \(\cos(kx) = \frac{1}{2}(e^{ikx} + e^{-ikx})\). (c) requires the product rule for differentiation. An eigenfunction is one for which \(\hat{p}\psi = (\text{constant})\cdot\psi\).

→ View solution

B-3. Normalize the wave function (where is a constant). That is

Normalize the wave function \(\Psi(x) = A e^{-x^2/(2a^2)}\) (where \(a > 0\) is a constant). That is, find the real positive constant \(A\) that satisfies

\[\int_{-\infty}^{+\infty}|\Psi(x)|^2\,dx = 1\]

You may use the Gaussian integral \(\int_{-\infty}^{+\infty}e^{-\alpha x^2}dx = \sqrt{\pi/\alpha}\) (\(\alpha > 0\)).

Hint

Integrate \(|\Psi|^2 = A^2 e^{-x^2/a^2}\). Set \(\alpha = 1/a^2\) and apply the Gaussian integral formula.

→ View solution

B-4. Properties of the Dirac Delta Function

Properties of the Dirac Delta Function

Using the property of the Dirac delta function

\[\int_{-\infty}^{+\infty} f(x)\,\delta(x - x_0)\,dx = f(x_0)\]

calculate the following.

(a) \(\int_{-\infty}^{+\infty} (3x^2 + 2)\,\delta(x - 1)\,dx\)

(b) \(\int_{-\infty}^{+\infty} e^{ikx}\,\delta(x)\,dx\)

(c) \(\int_{-\infty}^{+\infty} \psi^*(x)\,\delta(x - x')\,dx\) (where \(\psi(x)\) is an arbitrary wave function)

Hint

Directly apply the "sifting" property of the delta function. When \(\delta(x - x_0)\) appears in the integrand, it picks out the value of \(f(x)\) at \(x = x_0\).

→ View solution

B-5. For the stationary state wave function, calculate the probability density and show that it is time-independent

For the stationary state wave function \(\Psi(x,t) = \psi(x)\,e^{-iEt/\hbar}\), calculate the probability density \(|\Psi(x,t)|^2\) and show that it is time-independent.

Hint

Calculate \(|\Psi|^2 = \Psi^*\Psi\). Use the fact that the complex conjugate of \(e^{-iEt/\hbar}\) is \(e^{+iEt/\hbar}\).

→ View solution

B-6. Hamiltonian Operator

Apply the Hamiltonian operator

\[\hat{H} = -\frac{\hbar^2}{2m}\frac{d^2}{dx^2} + V(x)\]

to the wave function \(\psi(x) = Be^{-\kappa x}\) (for \(x > 0\), where \(\kappa > 0\) is a constant). Assume \(V(x) = 0\) (for \(x > 0\)). Show that \(\psi\) is an eigenfunction of \(\hat{H}\), and express the corresponding energy eigenvalue \(E\) in terms of \(\kappa, \hbar, m\).

Hint

Compute \(\frac{d^2}{dx^2}e^{-\kappa x} = \kappa^2 e^{-\kappa x}\) and verify that the result takes the form \(\hat{H}\psi = E\psi\).

→ View solution

B-7. Superposition of Two Energy Eigenstates

Superposition of Two Energy Eigenstates

For the superposition

\[\Psi(x,t) = \frac{1}{\sqrt{2}}\psi_1(x)\,e^{-iE_1 t/\hbar} + \frac{1}{\sqrt{2}}\psi_2(x)\,e^{-iE_2 t/\hbar}\]

calculate the probability density \(|\Psi(x,t)|^2\) and determine the angular frequency of oscillation of the interference term. Assume that \(\psi_1(x)\) and \(\psi_2(x)\) are real functions.

Hint

Expand \(|\Psi|^2 = \Psi^*\Psi\). The cross terms contain \(e^{\pm i(E_2 - E_1)t/\hbar}\). If \(\psi_1, \psi_2\) are real, you can use \(\psi_n^* = \psi_n\).

→ View solution

B-8. For the momentum operator, write down explicitly. Furthermore, apply to the wave function and find the result.

For the momentum operator \(\hat{p} = -i\hbar\frac{\partial}{\partial x}\), write down \(\hat{p}^2\) explicitly. Furthermore, apply \(\hat{p}^2\) to the wave function \(\psi(x) = A\sin(3\pi x/L)\) and find the result.

Hint

\(\hat{p}^2 = \hat{p}\cdot\hat{p} = \left(-i\hbar\frac{\partial}{\partial x}\right)^2 = -\hbar^2\frac{\partial^2}{\partial x^2}\). Compute the second derivative of the \(\sin\) function.

→ View solution

Medium

M-1. Derivation of the Time-Independent Schrödinger Equation via Separation of Variables

Derivation of the Time-Independent Schrödinger Equation via Separation of Variables

Assume the wave function takes the form \(\Psi(x,t) = \psi(x)\,T(t)\), and substitute it into the general Schrödinger equation (7.13):

\[i\hbar\frac{\partial\Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2\Psi}{\partial x^2} + V(x)\Psi\]

Divide both sides by \(\psi(x)\,T(t)\) to obtain the form "a function of \(x\) only \(=\) a function of \(t\) only," and show that both sides must be equal to a constant (call it \(E\)). Write down the two resulting ordinary differential equations, and solve the equation for \(T(t)\) to derive \(T(t) = e^{-iEt/\hbar}\).

Hint

After substitution, the left-hand side becomes \(i\hbar\psi(x)\frac{dT}{dt}\), and the right-hand side becomes \(\left[-\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi\right]T(t)\). Dividing both sides by \(\psi T\), the left-hand side becomes a function of \(t\) only, and the right-hand side becomes a function of \(x\) only. If functions of different independent variables are equal, then both must be constant (the separation of variables argument).

→ View solution

M-2. Derivation of Conservation of Probability (Continuity Equation)

Derivation of Conservation of Probability (Continuity Equation)

For the probability density \(\rho(x,t) = |\Psi(x,t)|^2\) and the probability current density

\[j(x,t) = \frac{\hbar}{2mi}\left(\Psi^*\frac{\partial\Psi}{\partial x} - \frac{\partial\Psi^*}{\partial x}\Psi\right)\]

derive the continuity equation

\[\frac{\partial\rho}{\partial t} + \frac{\partial j}{\partial x} = 0\]

using the Schrödinger equation (7.13). Furthermore, show that when \(\Psi\) approaches 0 sufficiently rapidly as \(x \to \pm\infty\), the total probability \(\int_{-\infty}^{+\infty}\rho\,dx\) is independent of time.

Hint

Refer to the derivation of equations (7.23)–(7.27) in the text. Compute \(\frac{\partial\rho}{\partial t}\), confirm that the terms involving \(V\) cancel, and verify that the remaining terms can be written as \(-\frac{\partial j}{\partial x}\). The time derivative of the total probability equals the boundary values of \(j\), namely \(j(+\infty) - j(-\infty)\), which vanishes if the wave function goes to 0 at infinity.

→ View solution

M-3. Calculation of Probability Current Density

Calculation of Probability Current Density

For the wave function \(\Psi(x,t) = Ae^{i(kx - \omega t)}\) (where \(A, k, \omega\) are real constants):

(a) Find the probability density \(\rho = |\Psi|^2\).

(b) Calculate the probability current density \(j = \frac{\hbar}{2mi}\left(\Psi^*\frac{\partial\Psi}{\partial x} - \frac{\partial\Psi^*}{\partial x}\Psi\right)\).

(c) Confirm that the obtained \(j\) can be written in the form \(j = \rho v\) using the particle velocity \(v = p/m = \hbar k/m\) and the probability density \(\rho\).

(d) Verify that the continuity equation \(\frac{\partial\rho}{\partial t} + \frac{\partial j}{\partial x} = 0\) is satisfied.

Hint

\(\Psi^* = Ae^{-i(kx-\omega t)}\) (when \(A\) is real). Use \(\frac{\partial\Psi}{\partial x} = ik\Psi\) and \(\frac{\partial\Psi^*}{\partial x} = -ik\Psi^*\).

→ View solution

M-4. Calculation of the Commutation Relation of Operators

Calculation of the commutation relation \([\hat{x}, \hat{p}]\) of operators

In the position representation, \(\hat{x} = x\) (multiplication) and \(\hat{p} = -i\hbar\frac{\partial}{\partial x}\). Calculate the commutator

\[[\hat{x}, \hat{p}] \equiv \hat{x}\hat{p} - \hat{p}\hat{x}\]

by acting on an arbitrary test function \(f(x)\), and show that

\[[\hat{x}, \hat{p}] = i\hbar\]
Hint

Calculate \(\hat{x}\hat{p}f(x) = x\left(-i\hbar\frac{df}{dx}\right)\) and \(\hat{p}\hat{x}f(x) = -i\hbar\frac{d}{dx}(xf)\) separately, then take the difference. Use the product rule \(\frac{d}{dx}(xf) = f + x\frac{df}{dx}\).

→ View solution

Advanced

A-1. Time Evolution of a Gaussian Wave Packet and Wave Packet Spreading

Time Evolution of a Gaussian Wave Packet and Wave Packet Spreading

The initial wave function of a free particle (\(V = 0\)) is given by

\[\Psi(x, 0) = \left(\frac{1}{2\pi\sigma_0^2}\right)^{1/4}\exp\left(-\frac{x^2}{4\sigma_0^2}\right)\]

where \(\sigma_0 > 0\) is a parameter characterizing the initial spread in position.

(a) Find the momentum-space amplitude \(\phi(k)\) by Fourier transforming this wave function:

\[\phi(k) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\Psi(x,0)\,e^{-ikx}\,dx\]

(b) Using the free particle dispersion relation \(\omega = \hbar k^2/(2m)\), compute the wave function at time \(t\):

\[\Psi(x,t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{+\infty}\phi(k)\,e^{i(kx - \omega t)}\,dk\]

and show that \(|\Psi(x,t)|^2\) remains a Gaussian distribution.

(c) Find the width (standard deviation) \(\sigma(t)\) of the probability density, and show that

\[\sigma(t) = \sigma_0\sqrt{1 + \left(\frac{\hbar t}{2m\sigma_0^2}\right)^2}\]

Discuss the physical meaning of the wave packet spreading over time from the perspective of the uncertainty principle.

Hint

(a) The Fourier transform of a Gaussian function is a Gaussian function. Use \(\int_{-\infty}^{+\infty}e^{-ax^2 + bx}dx = \sqrt{\pi/a}\,e^{b^2/(4a)}\) (for \(\text{Re}(a) > 0\)). (b) Perform the Gaussian integral again. Complete the square in the exponent for the \(k\) integration. Pay attention to the complex parameters. (c) Read off the variance of the Gaussian distribution from \(|\Psi(x,t)|^2\). Relate it to how the initial momentum uncertainty \(\Delta p \sim \hbar/(2\sigma_0)\) gives rise to the spreading in position.

→ View solution

A-2. Ehrenfest's Theorem

Ehrenfest's Theorem

For a particle in a general potential \(V(x)\), compute the time derivatives of the expectation values of position and momentum

\[\langle x \rangle = \int_{-\infty}^{+\infty}\Psi^* x\,\Psi\,dx, \quad \langle p \rangle = \int_{-\infty}^{+\infty}\Psi^*\left(-i\hbar\frac{\partial}{\partial x}\right)\Psi\,dx\]

using the Schrödinger equation, and derive the following Ehrenfest's theorem:

\[\frac{d\langle x\rangle}{dt} = \frac{\langle p\rangle}{m} \tag{i}\]
\[\frac{d\langle p\rangle}{dt} = -\left\langle\frac{dV}{dx}\right\rangle \tag{ii}\]

Furthermore, discuss how these results constitute the quantum mechanical counterpart of Newton's equation of motion \(F = ma\). In particular, explain why the expectation values exactly coincide with classical trajectories when \(V(x)\) is a polynomial of degree two or less in \(x\).

Hint

(i) Compute \(\frac{d\langle x\rangle}{dt} = \int \frac{\partial}{\partial t}(\Psi^* x \Psi)\,dx\). Replace \(\partial\Psi/\partial t\) and \(\partial\Psi^*/\partial t\) using the Schrödinger equation, then perform integration by parts. (ii) Similarly compute \(\frac{d\langle p\rangle}{dt}\). The key is the commutation relation between \(\hat{p}\) and \(V(x)\): \([\hat{p}, V(x)] = -i\hbar\frac{dV}{dx}\). Verify that if \(V\) is of degree two or less, then \(\langle dV/dx \rangle = \frac{dV}{dx}\big|_{x=\langle x\rangle}\) holds.

→ View solution