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Chapter 5: Problem 30

A one-dimensional harmonic oscillator is in its ground state for \(t<0\). For \(t \geq 0\) it is subjected to a time-dependent but spatially uniform force (not potential!) in the \(x\)-direction, $$ F(t)=F_{0} e^{-t / \tau} $$ a. Using time-dependent perturbation theory to first order, obtain the probability of finding the oscillator in its first excited state for \(t>0\). Show that the \(t \rightarrow \infty\) ( \(\tau\) finite) limit of your expression is independent of time. Is this reasonable or surprising? b. Can we find higher excited states?

Short Answer

Expert verified
The probability of finding the oscillator in the first excited state becomes constant as \( t \rightarrow \infty \). This is reasonable as the force diminishes. Higher states are possible but less probable.

Step by step solution

01

Set up the Hamiltonian

The Hamiltonian for a harmonic oscillator subjected to a force can be expressed as: \( H(t) = H_0 - xF(t) \), where \( H_0 \) is the unperturbed Hamiltonian of the harmonic oscillator \( \left( H_0 = \frac{p^2}{2m} + \frac{1}{2}m\omega^2 x^2 \right) \). The perturbation \( V(t) = -xF(t) = -xF_0 e^{-t/\tau} \).
02

Use time-dependent perturbation theory

According to first-order time-dependent perturbation theory, the transition amplitude \( c_n(t) \) to the \( n \)-th excited state is: \( c_n(t) = -\frac{i}{\hbar} \int_0^t \langle n | V(t') | 0 \rangle e^{i \omega_{n0} t'} dt' \), where \( \omega_{n0} = (E_n - E_0)/\hbar \) and \( E_n = \hbar \omega \left( n + \frac{1}{2} \right) \).
03

Calculate \( \langle 1 | V(t') | 0 \rangle \)

The matrix element for transition to the first excited state is \( \langle 1 | -xF_0 e^{-t'/\tau} | 0 \rangle = -F_0 e^{-t'/\tau} \langle 1 | x | 0 \rangle \). The position matrix element \( \langle 1 | x | 0 \rangle \) for a harmonic oscillator is known to be \( \sqrt{\frac{\hbar}{2m\omega}} \).
04

Integrate to find \( c_1(t) \)

Substitute the known values into the integral for \( c_1(t) \): \[ c_1(t) = -\frac{iF_0}{\hbar} \sqrt{\frac{\hbar}{2m\omega}} \int_0^t e^{(i\omega - 1/\tau)t'} dt' \] Evaluating this integral gives: \[ c_1(t) = \frac{F_0}{\hbar} \sqrt{\frac{\hbar}{2m\omega}} \frac{\tau}{1 - i\omega\tau} \left( 1 - e^{(i\omega - 1/\tau)t} \right) \].
05

Compute the probability

The probability \( P_1(t) \) of finding the oscillator in the first excited state is the modulus squared of \( c_1(t) \): \[ P_1(t) = \left| \frac{F_0}{\hbar} \sqrt{\frac{\hbar}{2m\omega}} \frac{\tau}{1 - i\omega\tau} \left( 1 - e^{(i\omega - 1/\tau)t} \right) \right|^2 \] In the limit as \( t \rightarrow \infty \), the exponential term vanishes, making \( P_1(\infty) = \left| \frac{F_0 \tau}{\hbar} \sqrt{\frac{\hbar}{2m\omega}} \right|^2 \cdot \frac{1}{1+\omega^2\tau^2} \).
06

Interpret the result

For large \( t \), the probability is indeed independent of \( t \), showing a constant result. This is reasonable because as time progresses, the external force contribution diminishes, reaching a steady state determined by the initial conditions and system properties.
07

Consider higher excited states

Higher excited states can also be reached through similar processes. However, the transition probabilities decrease significantly with higher energy levels due to increasingly smaller matrix elements like \( \langle n | x | 0 \rangle \) as \( n \) increases.

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Key Concepts

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

Harmonic Oscillator
A harmonic oscillator is a system in physics that undergoes periodic motion. Think of a mass attached to a spring. When disturbed from its equilibrium position, it experiences a restoring force proportional to the displacement.

The harmonic oscillator in quantum mechanics describes such a system at a quantum level, predicting phenomena like quantized energy levels. Its energy is given by the formula:
  • \( E_n = \hbar \omega \left( n + \frac{1}{2} \right) \)
where \(\hbar\) is the reduced Planck's constant, \(\omega\) is the angular frequency, and \(n\) is an integer denoting the energy level.

These levels are discrete, meaning the oscillator can only possess certain energies, unlike classical systems where any energy is possible.
Quantum Mechanics
Quantum Mechanics is the branch of physics that deals with the behavior of matter and light on small scales鈥攍ike atoms and subatomic particles.

Instead of definite paths and variables, quantum mechanics uses probabilities to predict the behavior of systems. For instance, an electron does not have a definite position, but a probability distribution showing where it might be found.

One key aspect of quantum mechanics is wavefunction, which encodes all the information about the state of a quantum system. The time-dependent Schr枚dinger equation governs how these wavefunctions evolve over time.
Excited States
In quantum mechanics, an excited state of a system is any energy state higher than the ground state.
  • The ground state is the lowest energy state, where a particle naturally resides.
  • When energy is added to a system, it can move to an excited state.
These states are significant as they influence how atoms and molecules interact with each other. For example, when electrons in an atom absorb energy, they jump to higher energy levels, or excited states. These states are temporary, and the electrons will return to the ground state, often releasing energy as light. When dealing with a harmonic oscillator, understanding excited states is crucial for analyzing transitions between energy levels.
Matrix Elements
Matrix elements in quantum mechanics are central to determining the likelihood of transitions between states.

For any two states \(|m\rangle\) and \(|n\rangle\), the matrix element \(\langle m | \hat{O} | n \rangle\) represents the expected value of an observable \(\hat{O}\) when transitioning between these states.

In our example, we computed the matrix element for the transition to the first excited state \(\langle 1 | x | 0 \rangle\). This element is crucial for calculating transition probabilities using time-dependent perturbation theory.

Matrix elements effectively link observables like position or momentum to probability amplitudes, and help describe how quantum states change over time during such transitions.

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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) \(2246 .\)

Estimate the ground-state energy of a one-dimensional simple harmonic oscillator using $$ \langle x \mid \tilde{0}\rangle=e^{-\beta|x|} $$ as a trial function with \(\beta\) to be varied.

Consider a one-dimensional simple harmonic oscillator whose classical angular frequency is \(\omega_{0}\). For \(t<0\) it is known to be in the ground state. For \(t>0\) there is also a time-dependent potential $$ V(t)=F_{0} x \cos \omega t $$ where \(F_{0}\) is constant in both space and time. Obtain an expression for the expectation value \(\langle x\rangle\) as a function of time using time- dependent perturbation theory to lowest nonvanishing order. Is this procedure valid for \(\omega \simeq \omega_{0} ?\)

A particle of mass \(m\) moves in a potential well \(V(x)=m \omega^{2} x^{2} / 2\). Treating relativistic effects to order \(\beta^{2}=(p / m c)^{2}\), find the ground-state energy shift.

Consider an atom made up of an electron and a singly charged \((Z=1)\) triton \(\left({ }^{3} \mathrm{H}\right)\). Initially the system is in its ground state \((n=1, l=0)\). Suppose the system undergoes beta decay, in which the nuclear charge suddenly increases by one unit (realistically by emitting an electron and an antineutrino). This means that the tritium nucleus (called a "triton") turns into a helium \((Z=2)\) nucleus of mass \(3\left({ }^{3} \mathrm{He}\right)\). a. Obtain the probability for the system to be found in the ground state of the resulting helium ion. b. The available energy in tritium beta decay is about \(18 \mathrm{keV}\) and the size of the \({ }^{3} \mathrm{He}\) atom is about \(1 \AA\). Check that the time scale \(T\) for the transformation satisfies the criterion of validity for the sudden approximation.
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