# Assignment 1: Math Foundations of Quantum Mechanics

CHEM106: Assignment 6

Harmonic Oscillator

1.For the harmonic oscillator system, the solution of the Schrödinger equation leads to the quantized energy .

A.Define the zero-point energy of the system.

The zero-point energy (ZPE) for a quantum mechanical system is defined as the lowest possible energy allowed. In the case of the harmonic oscillator, the energy depends on the quantum number v, and v = 0, 1, 2, 3, ….

Thus,

B.Determine the energy gap between state v + 1 and state v.

The energy gap between the state v + 1 and v can be calculated as

.

It is worth noting that the energy gap is related to the frequency of molecular vibration (see Unit 7, Section 3 later).

2.For the harmonic oscillator, the solution of the Schrödinger equation leads to the ground state wave function .

- Show that the wave function is normalized.

We need to evaluate the integral . If M = 1, then the wave function is normalized.

Look up the integration table, and find that .

We have .

Thus, is normalized.

B.Calculate the average value of the linear momentum.

According to the average value postulate of quantum mechanics, the expectation value for any physical observable A is defined as

,

where is the complex conjugate of , and the integration is done over the entire space.

Thus, the average linear momentum can be evaluated by the following integral:

Consider that is an odd function, and its integral over the whole range must vanish, i.e., .

Thus, we obtain.

3.Tunneling occurs in the quantum harmonic oscillator. For a classical harmonic oscillator, the particle cannot go beyond the classical energy barriers (i.e., points where the total energy equals the potential energy).

- Identify these points for a quantum-mechanical harmonic oscillator in its ground state.

For the ground state of the harmonic oscillator, the total energy is Panel (a) of the above figure plots the potential energy as a function of displacement (x). At the classical barrier (position a and –a), the total energy of the system is all in the form of potential energy. That is .

Solve for a, and we obtain

.

For the harmonic oscillator, the vibrational frequency is given by .

Plug into the above equation, and we obtain

.

- Set up an integral defining the probability of tunneling (i.e., the probability that the particle will go beyond the classical barriers). [You do not need to evaluate the integral.]

The ground state wave function for the harmonic oscillator is plotted in panel (b) of the above figure. For the +x direction, the probability of tunneling can be evaluated as

.

Due to the symmetry, the overall probability of tunneling should be twice of that of one (+) direction. Thus, the overall probability of tunneling is

,

where .

4.Vibration of a diatomic molecule can be modeled as a harmonic oscillator. Calculate the ratio of the vibrational frequency between hydrogen chloride (H35Cl) and deuterium chloride (D35Cl).

For the harmonic oscillator, the vibrational frequency is given by ,

where k is the force constant, and is the reduced mass defined by.

The frequency for H35Cl and D35Cl are defined by and , respectively.

For various isotopes of the same element, the force constant k stays the same. We have . Therefore, the frequency ratio is

.