Chapter 8, Variational Methods and Simple Perturbation Theory Video Solutions, Quantum mechanics

Problem 1

Apply the variational method to estimate the ground state energy of a particle confined in a one-dimensional box for which $V=0$ for $-a<x<a$, and $\psi( \pm a)=0$.
(a) First, use an unnormalized trapezoidal trial function which vanishes at $\pm a$ and is symmetric with respect to the center of the well:
$$
\psi_t(x)= \begin{cases}(a-|x|) & b \leq|x| \leq a \\ a-b & |x| \leq b\end{cases}
$$

Try the choice $b=0$ (triangular trial function) and then improve on this by optimizing the parameter $b$.
(b) A more sophisticated trial function is parabolic, again vanishing at the endpoints and even in $x$.
(c) Use a quartic trial function of the form
$$
\psi_i(x)=\left(a^2-x^2\right)\left(\alpha x^2+\beta\right)
$$
where the ratio of the adjustable parameters $\alpha$ and $\beta$ is determined variationally.
(d) Compare the results of the different variational calculations with the exact ground state energy, and, using normalized wave functions, evaluate the meansquare deviation $\int_{-a}^a\left|\psi(x)-\psi_t(x)\right|^2 d x$ for the various cases.
(e) Show that the variational procedure produces, in addition to the approximation to the ground state, an optimal quartic trial function with nodes between the endpoints. Interpret the corresponding stationary energy value. ${ }^7$

Check back soon!

Problem 2

Using scaled variables, as in Section 5.1, consider the anharmonic oscillator Hamiltonian,
$$
H=\frac{1}{2} p_{\xi}^2+\frac{1}{2} \xi^2+\lambda \xi^4
$$
where $\lambda$ is a real-valued parameter.
(a) Estimate the ground state energy by a variational calculation, using as a trial function the ground state wave function for the harmonic oscillator
$$
H_0(\omega)=\frac{1}{2} p_{\xi}^2+\frac{1}{2} \omega^2 \xi^2
$$
where $\omega$ is an adjustable variational parameter. Derive an equation that relates $\omega$ and $\lambda$.
(b) Compute the variational estimate of the ground state energy of $H$ for various positive values of the strength $\lambda$.
(c) Note that the method yields answers for a discrete energy eigenstate even if $\lambda$ is slightly negative. Draw the potential energy curve to judge if this result makes physical sense. Explain.

Check back soon!

View

Problem 3

In first-order perturbation theory, calculate the change in the energy levels of a linear harmonic oscillator that is perturbed by a potential $g x^4$. For small values of the coefficient, compare the result with the variational calculation in Problem 2.

Andreas Papavassiliou

Numerade Educator

Problem 4

Using a Gaussian trial function, $e^{-\lambda x^2}$, with an adjustable parameter, make a variational estimate of the ground state energy for a particle in a Gaussian potential well, represented by the Hamiltonian
$$
H=\frac{p^2}{2 m}-V_0 e^{-\alpha x^2} \quad\left(V_0 \text { and } \alpha>0\right)
$$

Check back soon!

Problem 5

Show that as inadequate a variational trial function as
$$
\psi(x)=\left\{\begin{array}{cc}
C\left(1-\frac{|x|}{a}\right) & \text { for }|x| \leq a \\
0 & \text { for }|x|>a
\end{array}\right.
$$
yields, for the optimum value of $a$, an upper limit to the ground state energy of the linear harmonic oscillator, which lies within less than 10 percent of the exact value.

Check back soon!

20:07

Problem 6

A particle of mass $m$ moves in a potential $V(\mathbf{r})$. The $n$-th discrete energy eigenfunction of this system, $\psi_n(\mathbf{r})$, corresponds to the energy eigenvalue $E_n$. Apply the variational principle by using as a trial function,
$$
\psi_t(\mathbf{r})=\psi_n(\lambda \mathbf{r})
$$
where $\lambda$ is a variational (scaling) parameter, and derive the virial theorem for stationary states.

Mahnoor Amin

Numerade Educator

02:02

Problem 7

. In Chapter 6 it was shown that every one-dimensional square well supports at least one bound state. By use of the variational principle, prove that the same is true for any one-dimensional potential that is negative for all values of $x$ and that behaves as $V \rightarrow 0$ as $x \rightarrow \pm \infty$.

Kratika Bhadauria

Numerade Educator

04:19

Problem 8

Work out an approximation to the energy splitting between the second and third excited levels of the double oscillator defined in Section 8.5, assuming the distance between the wells to be very large compared with the classical amplitude of the zero-point vibrations.

Mahnoor Amin

Numerade Educator

09:26

Problem 9

Solve the energy eigenvalue problem for a particle that is confined in a two-dimensional square box whose sides have length $L$ and are oriented along the $x$ - and $y$-coordinate axes with one corner at the origin. Find the eigenvalues and eigenfunctions, and calculate the number of eigenstates per unit energy interval for high energies.
A small perturbation $V=C x y$ is now introduced. Find the approximate energy change of the ground state and the splitting of the first excited energy level. For the given perturbation, construct the optimal superpositions of the unperturbed wave functions in the case of the first excited state.

Noor Aldeen Almusleh

Numerade Educator

10:01

Problem 10

. As an example of Problem 2 in Chapter 7, apply the WKB approximation to the double harmonic oscillator of Section 8.5, and contrast the energy level splitting of the two lowest levels with the results obtained in Section 8.5.

Guilherme Barros

Numerade Educator

02:29

Problem 11

The energy $E_0(a)$ of the lowest eigenstate of a double harmonic oscillator with fixed $\omega$ depends on the distance $a$ and has a minimum at $a=a_0$ (see Figure 8.5). Adapt the Hellmann-Feynman theorem for the expectation value of a parameter-dependent Hamiltonian (Exercise 8.30) to this problem, and show that if $a=a_0$, the expectation value of $|x|$ is equal to $a_0$.

Lottie Adams

Numerade Educator

10:01

Problem 12

Apply the WKB approximation to a periodic potential in one dimension, and derive an implicit equation for the dispersion function $E(k)$. Estimate the width of the valence band of allowed bound energy levels.

Guilherme Barros

Numerade Educator

Problem 13

Assume that $n$ unperturbed, but not necessarily degenerate, eigenstates $|k\rangle$ of an unperturbed Hamiltonian $H_0$ (with $k=1,2, \ldots, n$ ) all interact with one of them, say $|1\rangle$, but not otherwise so that the perturbation matrix elements $\left\langle k|V| k^{\prime}\right\rangle \neq 0$ only if either $k=k^{\prime}$ or $k=1$ or $k^{\prime}=1$. Solve the eigenvalue problem in the $n$-dimensional vector space exactly and derive an implicit equation for the perturbed energies. Using a graphic method, discuss the solutions of the eigenvalue problem for various assumed values of the nonvanishing matrix elements of $V$, and exhibit the nature of the perturbed eigenstates.

Check back soon!