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
.