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Chapter 8: Problem 18

(a) Calculate the rate for spontaneous transitions between the \(n=1\) and \(n=0\) states of a simple harmonic oscillator, carrying charge \(e\). Take the mass of the oscillator to be equal to the mass of an atom of some typical ionic molecule, and the restoring force constant \(C\) to be \(10^{3}\) joules \(/ \mathrm{m}^{2}\), which is typical for such a molecule. (Hint: Normalized eigenfunctions must be used.) (b) From the transition rate, estimate the average time required to complete the transition. This is the lifetime of the \(n=1\) vibrational state of the molecule.

Short Answer

Expert verified
The calculation of the transition rate involves calculation of matrix elements of the interaction Hamiltonian using the wave functions for harmonic oscillator states. The average transition time can be obtained as the inverse of calculated transition rate.

Step by step solution

01

Set Up the Problem

Identify the important variables. The mass of the atom (m) is not given but is assumed to be that of a typical atom in an ionic molecule. The oscillation constant (C) is given as \(10^{3}\) joules/m\(^{2}\). The charge (e) is the charge of a proton. The transition is from the \(n=1\) state to the \(n=0\) state.
02

Calculate Transition Rate

The spontaneous transition rate for an atom in an ionic molecule can be calculated using Fermi's Golden Rule. The result of this computation gives us the rate of transitions per unit time and can be represented as: \(W = \frac{2\pi}{\hbar}|<1|He_{int}|0>|^{2} \rho\). Here, \(\rho\) is the density of states, \(\hbar\) is the reduced Planck constant, and \(He_{int}\) is the interaction Hamiltonian between the states. In the case of a harmonic oscillator, \(He_{int}\) takes the form of \(eEz\), where E is the electric field and z is the position operator. Using the wave functions for harmonic oscillators, the matrix elements of \(He_{int}\) can be computed. This requires the knowledge of the normalized wave functions for the harmonic oscillator.
03

Estimate Average Transition Time

The average time required for the completion of a transition, or 'lifetime' of a state, is inverse of the transition rate: \(\tau = \frac{1}{W}\). Once you calculate the transition rate in the previous step, you can then calculate the average transition time.

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Key Concepts

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

Spontaneous Transition
In quantum mechanics, a spontaneous transition refers to the process by which a quantum system moves from a higher energy state to a lower one without any external influence. This is a fundamental concept observed in various systems, including atomic and molecular structures, like our harmonic oscillator. During a spontaneous transition, the system naturally shifts to a more stable, lower energy state, releasing energy in the form of radiation.
Understanding spontaneous transitions helps to explain many phenomena in physics and chemistry.
  • These transitions are stochastic, meaning they occur randomly over time.
  • The probability of such transitions is governed by quantum rules rather than classical physics.
  • The energy difference between the states determines the frequency of emitted radiation.
Thus, spontaneous transitions are essential for explaining how systems reach their ground state, shedding light on everything from laser operations to stellar lifecycles.
Fermi's Golden Rule
Fermi's Golden Rule is a critical principle used to calculate the transition rate per unit time for spontaneous transitions between quantum states. This rule is indispensable in predicting how often such transitions will occur. It is expressed using the formula:
\(W = \frac{2\pi}{\hbar}|<1|He_{int}|0>|^{2} \rho\)

  • Here, \(W\) represents the transition rate.
  • \(\hbar\) is the reduced Planck constant, a fundamental constant in quantum mechanics.
  • The term \(\rho\) refers to the density of states, representing how many states are available at a particular energy level.
  • \(<1|He_{int}|0>\) is the matrix element that quantifies the interaction between initial and final states.
This formula helps quantify the spontaneous transition's likelihood, making it a cornerstone in both theoretical and applied physics. Its applications include understanding nuclear decay, electromagnetic transitions in atoms, and even quantum computing processes.
Harmonic Oscillator Wave Functions
The harmonic oscillator is a model used to describe particles in a potential well, somewhat akin to a mass on a spring. It鈥檚 used in quantum mechanics to illustrate particles like atoms and molecules in vibrational states. Harmonic oscillator wave functions are solutions to the Schr枚dinger equation for these systems, enabling calculations of quantum state transitions.
  • These wave functions are characterized by quantum numbers which define their energy and shape.
  • In our context, they help determine the matrix elements necessary for applying Fermi's Golden Rule.
  • Each state is described by Hermite polynomials with a Gaussian envelope.
The wave functions are normalized, meaning their total probability sums to one over all space.
They differ from classical systems, as quantum harmonic oscillators allow only discrete energy levels. This quantization explains why transitions between these prescribed states require specific energies, and these calculations are critical for understanding phenomena like vibrational spectroscopy in molecules.

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Most popular questions from this chapter

$$ \text { Enumerate the possible values of } j \text { and } m_{i} \text { for states in which } l=3 \text { and } s=1 / 2 \text {. } $$

Evaluate the magnetic field produced by a circular current loop at a point on the axis of symmetry far from the loop. Then evaluate the magnetic field produced at the same point by a dipole formed from two separated magnetic monopoles located at the center of the loop and lying along the axis of symmetry. Show that the fields are the same if the current in the loop and its area are related to the magnetic moment of the dipole by (8-2). Can you see how to extend the argument to show that the fields will be the same at all points far from the loop or dipole, and independent of the shape of the loop?

A beam of hydrogen atoms in their ground state is sent through a Stern-Gerlach magnet, which splits it into two components according to the two spin orientations. One component is stopped by a diaphragm at the end of the magnet, and the other continues into a second Stern-Gerlach magnet which is coaxial with the beam leaving the first magnet, but is rotated relative to the first magnet about their approximately common axes through an angle \(\alpha\). There is a second diaphragm fixed on the end of the second magnet which also allows only one component to pass. Describe qualitatively how the intensity of the beam passing the second diaphragm depends on \(\alpha\).

Determine the field gradient of a \(50 \mathrm{~cm}\) long Stern-Gerlach magnet that would produce a \(1 \mathrm{~mm}\) separation at the end of the magnet between the two components of a beam of silver atoms emitted with typical kinetic energy from a \(960^{\circ} \mathrm{C}\) oven. The magnetic dipole moment of silver is due to a single \(l=0\) electron, just as for hydrogen.

The relativistic shift in the energy levels of a hydrogen atom due to the relativistic dependence of mass on velocity can be determined by using the atomic eigenfunctions to calculate the expectation value \(\overline{\Delta E}_{\mathrm{rel}}\) of the quantity \(\Delta E_{\mathrm{rel}}=E_{\mathrm{rel}}-E_{\text {clas }}\), the difference between the relativistic and classical expressions for the total energy \(E\). Show that for \(p\) not too large $$ \Delta E_{\mathrm{rel}} \simeq-\frac{p^{4}}{8 m^{3} c^{2}}=-\frac{E^{2}+V^{2}-2 E V}{2 m c^{2}} $$ so that $$ \begin{aligned} \overline{\Delta E}_{\mathrm{rel}}=&-\frac{E_{n}^{2}}{2 m c^{2}}-\frac{e^{4}}{\left(4 \pi \epsilon_{0}\right)^{2} 2 m c^{2}} \int \psi_{n l j m l}^{*} \frac{1}{r^{2}} \psi_{n l j m_{1}} d \tau \\ &-\frac{E_{n} e^{2}}{4 \pi \epsilon_{0} m c^{2}} \int \psi_{m l j m_{j}}^{*} \frac{1}{r} \psi_{n l j m_{j}} d \tau \end{aligned} $$
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