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Chapter 5: Problem 7

Consider an isotropic harmonic oscillator in two dimensions. The Hamiltonian is $$ H_{0}=\frac{p_{x}^{2}}{2 m}+\frac{p_{y}^{2}}{2 m}+\frac{m \omega^{2}}{2}\left(x^{2}+y^{2}\right) $$ a. What are the energies of the three lowest-lying states? Is there any degeneracy? b. We now apply a perturbation $$ V=\delta m \omega^{2} x y $$ where \(\delta\) is a dimensionless real number much smaller than unity. Find the zerothorder energy eigenket and the corresponding energy to first order [that is, the unperturbed energy obtained in (a) plus the first-order energy shift] for each of the three lowest-lying states. c. Solve the \(H_{0}+V\) problem exactly. Compare with the perturbation results obtained in (b).

Short Answer

Expert verified
The three lowest energies are \( \hbar\omega, 2\hbar\omega \) (degenerate). No first-order energy shift for states due to the perturbation.

Step by step solution

01

Understand the System

The isotropic harmonic oscillator in two dimensions is described by the Hamiltonian \( H_0 = \frac{p_x^2}{2m} + \frac{p_y^2}{2m} + \frac{m\omega^2}{2}(x^2 + y^2) \). The system has two degrees of freedom, leading to separable solutions for each dimension.
02

Determine Energy Levels

The energy levels of a 2D isotropic harmonic oscillator are given by \( E_{n_x, n_y} = \hbar \omega (n_x + n_y + 1) \), where \( n_x \) and \( n_y \) are quantum numbers for each dimension. The three lowest energy levels are:- \( E_{0,0} = \hbar \omega \times (0 + 0 + 1) = \hbar \omega \)- \( E_{1,0} = \hbar \omega \times (1 + 0 + 1) = 2 \hbar \omega \)- \( E_{0,1} = \hbar \omega \times (0 + 1 + 1) = 2 \hbar \omega \)The states \( E_{1,0} \) and \( E_{0,1} \) are degenerate.
03

Apply Perturbation and Find Energy Shift

Consider the perturbation \( V = \delta m \omega^2 x y \). For the zeroth-order energy eigenkets \(|n_x, n_y\rangle\), we use non-degenerate first-order perturbation theory where the first-order energy correction is given by \( \langle n_x, n_y | V | n_x, n_y \rangle \). For the states:- \( |0,0\rangle: \langle 0,0 | V | 0,0 \rangle = 0 \)- \( |1,0\rangle: \langle 1,0 | V | 1,0 \rangle = 0 \)- \( |0,1\rangle: \langle 0,1 | V | 0,1 \rangle = 0 \)There is no first-order energy shift because the expectation values are zero.
04

Solve the Perturbed Hamiltonian Exactly

Consider the combined Hamiltonian \( H_0 + V \). The perturbation \( V = \delta m \omega^2 x y \) is a cross term that doesn't affect energy eigenvalues in the lowest order for non-degenerate perturbation theory, but in exact diagonalization, it couples states like \(|1,0\rangle\) and \(|0,1\rangle\). Solve \([H_0 + V]\) for eigenvalues, confirming eigenenergies match the perturbative results.
05

Compare Exact and Perturbative Results

The exact diagonalization will show that the perturbation does not change the energy levels in the first order because all terms \( \langle n_x, n_y | V | n_x', n_y' \rangle \) are zero for these states. Thus, perturbational results are consistent with exact results for first-order shifts.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quantum Mechanics
Quantum mechanics is a fundamental theory in physics that describes the nature of physical systems at the smallest scales, such as atoms and subatomic particles. It relies on probabilities and uncertainties to predict the behavior of these systems rather than certainty or predetermined pathways.

In quantum mechanics, particles like electrons exhibit wave-like properties, which are described by wave functions. These wave functions provide the probability of finding the particle in a particular region of space.

One key aspect of quantum mechanics is that energy levels are quantized, meaning they can only take on specific discrete values. This is a stark contrast to classical physics, where energy levels form a continuum.

The isotropic harmonic oscillator is a pivotal concept in quantum mechanics, often used to model the behavior of particles in a potential field, like atoms in molecules or crystals. It plays a crucial role in explaining the quantized energy levels in two-dimensional systems.
Perturbation Theory
Perturbation theory is a mathematical technique used in quantum mechanics to find an approximate solution to a problem that cannot be solved exactly. It is highly useful when dealing with systems influenced by small disturbances or 'perturbations'.

Essentially, perturbation theory helps us understand how a system behaves when it is slightly changed by analyzing how small terms (perturbations) affect the main Hamiltonian of the system. It assumes that the perturbation is weak compared to the unperturbed system, allowing physicists to treat the effect as a correction to be applied to the known solutions.

In the context of the 2D isotropic harmonic oscillator, perturbation theory helps in understanding how the system's energy levels shift when a small cross-term perturbation, like \(V = \delta m \omega^2 x y\), is added to the Hamiltonian. This technique allows physicists to predict changes in energy levels due to this additional term, without needing to solve the problem exactly, as illustrated in the original exercise.
Two-Dimensional Systems
Two-dimensional systems refer to physical frameworks that are confined to two spatial dimensions. Such systems offer simplified models that are crucial for understanding complex phenomena in condensed matter physics and material science.

An isotropic 2D harmonic oscillator is a prime example of a two-dimensional system, where the system exhibits symmetry over its two dimensions 鈥� both dimensions behave identically. This makes the mathematical treatment more manageable and illustrates fundamental quantum principles in a controlled environment.

In quantum mechanics, these systems have two degrees of freedom, often denoted as \(x\) and \(y\), each contributing independently to the system's behavior. The energy levels of these systems depend on quantum numbers for each spatial dimension, creating rich structures like degeneracies, as seen in the isotropic harmonic oscillator.
Energy Levels
In quantum mechanics, energy levels refer to the discrete quantized states that a quantum system can occupy. These levels define how much energy a particle, such as an electron, can have while existing in a given state within a potential, like a harmonic oscillator.

The energy levels of a 2D isotropic harmonic oscillator are determined by quantum numbers \(n_x\) and \(n_y\), corresponding to oscillations in the \(x\) and \(y\) dimensions, respectively. The energies are given by the formula \(E_{n_x, n_y} = \hbar \omega (n_x + n_y + 1)\).

Degeneracy occurs when two or more different states have the same energy, as observed with \(E_{1,0}\) and \(E_{0,1}\) in the problem. These degeneracies play an important role when systems are exposed to perturbations, as they can influence how energy levels change.
Hamiltonian
The Hamiltonian in quantum mechanics is an operator corresponding to the total energy of the system. It includes both potential and kinetic energy terms and is central to solving quantum mechanical problems.

For the 2D isotropic harmonic oscillator, the Hamiltonian is given by \(H_{0} = \frac{p_{x}^{2}}{2 m} + \frac{p_{y}^{2}}{2 m} + \frac{m \omega^{2}}{2}(x^{2} + y^{2})\). It captures both the kinetic energy of the system's movement in the \(x\) and \(y\) directions and the potential energy due to the harmonic oscillator, which depends on the position.

When perturbations are introduced, the Hamiltonian is modified to include additional terms. This new Hamiltonian, \(H_{0} + V\), represents the modified total energy of the system and is used to study how the system's energy levels and behavior are affected by the perturbation.

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Most popular questions from this chapter

This problem highlights anomalies in the "exponential" decay of a state. It is inspired by Winter, Phys. Rev., 123 (1961) 1503 , but modern computer applications make it straightforward to directly evaluate the integrals numerically. Consider a particle of mass \(m\) that is initially inside a "well" bounded by an infinite wall to the left and a \(\delta\)-function potential on the right: The infinite wall is located at \(x=-a\), and the potential at \(x=0\) is \(U \delta(x)\) where \(U\) is a positive constant. The figure also shows a plausible "ground state" initial wave function \(\Psi(x, t=0)=(2 / a)^{1 / 2} \sin (n \pi x / a)\) with \(n=1\). a. Show that the wave function at all times can be written as $$ \Psi(x, t)=2 n\left(\frac{2}{a}\right)^{1 / 2} \int_{0}^{\infty} d q \frac{e^{-i T q^{2}} q \sin q[q \sin (l+1) q+f]}{\left(q^{2}-n^{2} \pi^{2}\right)\left(q^{2}+G q \sin 2 q+G^{2} \sin ^{2} q\right)} $$ where \(q \equiv\left[a(2 m E)^{1 / 2}\right] / \hbar\) for a particle with energy \(E, T \equiv \hbar t / 2 m a^{2}, l \equiv x / a\), and \(G \equiv 2 m a U / \hbar^{2}\) are all dimensionless quantities, and \(f=0\) for \(-a \leq x \leq 0\) and \(f=G \sin q \sin l q\) for \(x \geq 0\). This is most easily done by expanding the wave function in energy eigenstates \(|E\rangle\), as $$ \Psi(x, t)=\int_{0}^{\infty} d E \phi_{E}(x) e^{-i E t / \hbar} $$ where \(\phi_{E}(x)\) is an energy eigenfunction and \(\left\langle E \mid E^{\prime}\right\rangle=\delta\left(E-E^{\prime}\right)\). b. Write a computer program to (numerically) integrate the probability of finding the particle inside the well. Carry out the integration for a series of values of \(T\) between zero and 12 , using the same parameters as Winter, namely \(n=1\) and \(G=6\). Plotting these probabilities as a function of \(T\) should resemble Figure 2 of Winter's paper. Fit the points for \(2 \leq T \leq 8\) to an exponential, and compare the decay time to Winter's value of \(0.644\). c. Examine the behavior for \(T \geq 8\), and compare to the behavior Winter found for the current at \(x=0\). This suggests an experimental measurement. See Norman et al., Phys. Rev. Lett., \(\mathbf{6 0}\) (1988) \(2246 .\)

The unperturbed Hamiltonian of a two-state system is represented by $$ H_{0}=\left(\begin{array}{cc} E_{1}^{0} & 0 \\ 0 & E_{2}^{0} \end{array}\right) \text {. } $$ There is, in addition, a time-dependent perturbation $$ V(t)=\left(\begin{array}{cc} 0 & \lambda \cos \omega t \\ \lambda \cos \omega t & 0 \end{array}\right) \quad(\lambda \mathrm{real}) $$ a. At \(t=0\) the system is known to be in the first state, represented by $$ \left(\begin{array}{l} 1 \\ 0 \end{array}\right) $$ Using time-dependent perturbation theory and assuming that \(E_{1}^{0}-E_{2}^{0}\) is not close to \(\pm \hbar \omega\), derive an expression for the probability that the system be found in the second state represented by $$ \left(\begin{array}{l} 0 \\ 1 \end{array}\right) $$ as a function of \(t(t>0)\). b. Why is this procedure not valid when \(E_{1}^{0}-E_{2}^{0}\) is close to \(\pm \hbar \omega\) ?

Consider a composite system made up of two spin \(\frac{1}{2}\) objects. For \(t<0\), the Hamiltonian does not depend on spin and can be taken to be zero by suitably adjusting the energy scale. For \(t>0\), the Hamiltonian is given by $$ H=\left(\frac{4 \Delta}{\hbar^{2}}\right) \mathbf{S}_{1} \cdot \mathbf{S}_{2} $$ Suppose the system is in \(|+-\rangle\) for \(t \leq 0\). Find, as a function of time, the probability for being found in each of the following states \(|++\rangle,|+-\rangle,|-+\rangle\), and \(|--\rangle\). a. By solving the problem exactly. b. By solving the problem assuming the validity of first-order time-dependent perturbation theory with \(H\) as a perturbation switched on at \(t=0\). Under what condition does (b) give the correct results?

A particle of mass \(m\) constrained to move in one dimension is confined within \(0L \\ V=0 & \text { for } 0 \leq x \leq L \end{array} $$ Obtain an expression for the density of states (that is, the number of states per unit energy interval) for high energies as a function of \(E\). (Check your dimension!)

Compute the Stark effect for the \(2 s_{1 / 2}\) and \(2 p_{1 / 2}\) levels of hydrogen for a field \(\mathscr{E}\) sufficiently weak so that \(e \mathscr{E} a_{0}\) is small compared to the fine structure, but take the Lamb \(\operatorname{shift} \delta(\delta=1057 \mathrm{MHz})\) into account (that is, ignore \(2 p_{3 / 2}\) in this calculation). Show that for \(e \mathscr{E} a_{0} \ll \delta\), the energy shifts are quadratic in \(\mathscr{E}\), whereas for \(e \mathscr{E} a_{0} \gg \delta\) they are linear in \(\mathscr{E}\). Briefly discuss the consequences (if any) of time reversal for this problem. This problem is from Gottfried (1966), Problem 7-3.
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