Ch 9pages 465-469

Lecture 22 – Harmonic oscillator

A quantum mechanical particle is described through its wave function (x,t)

The square of the wave unction (x,t)2 characterizes the particle distribution in space and is a measure of the probability of finding a particle at a certain time and place

For example, the probability of finding a particle within a certain volume V is:

Summary of lecture 21

V

dxtx ),(2

The duality of matter introduces a probabilistic nature to measurements, so that we can calculate observables only in a probabilistic sense. This is done through the expectation value of a variable O, which can be calculated for a quantum system using the expression:

Summary of lecture 21

dxOO *

For example, the average position of an electron in a molecule is given by:

dxxx *

The wave function can be calculated by solving the Schroedinger Equation:

Summary of lecture 21

The time-independent component of the wave function (x) is obtained by solving the time-independent Schroedinger equation (when the potential is independent on time):

ttxihtxtxV

xtx

mh

),(

2),(),(),(

8 2

2

2

2

)()()()(8 2

2

2

2

xExxVxx

mh

We have calculated the wave function for a particle in a box:

Summary of lecture 21

And the corresponding energy levels:

Lxn

Lx sin2)(

E n hmLn 2 2

28

Mathematically, quantization results when particle waves are constrained in space by boundary conditions.

The Classical Linear Harmonic Oscillator

Classical mechanical problems are very often solved by introducing the so-called Hamiltonian, which is defined as:

The classical linear harmonic oscillator has the following Hamiltonian:

)(2

),(2

xVmppxH

22),(

22 xmppxH

Where the frequency of the oscillator is:

2

m

The Classical Linear Harmonic Oscillator

The trajectory x(t), p(t) of the oscillator is obtained by solving Hamilton’s equations of motion:

The solution is (homework) (A is the amplitude of the motion)

We can introduce a quantum mechanical Hamiltonian as:

dttdp

xpxH )(),(

dttdx

ppxH )(),(

tAtx cos)( tmAtp sin)(

)(8 2

2

2

2

xVdxd

mhH

The Time-Independent Schroedinger Equation can be written: )()( xExH

Quantum Mechanical Linear Harmonic Oscillator

The quantum mechanical treatment of the linear harmonic oscillator (LHO) is one of the most important applications of quantum mechanics

The LHO is used as a simple approximation to molecular bond vibrations and rotations, for example, and forms the basis of much spectroscopy.

The time-independent Schroedinger equation for the LHO is:

This equation may be rewritten as:

)()(28

2

2

2

2

2

xExxxm

h

)()(28

22

2

2

2

2

xExxmxm

h

)()()( 22

2

xqqqq

2 mh

q x

4 Eh

Quantum Mechanical Linear Harmonic Oscillator

Although there are no solid boundary conditions as there was with the particle in the box, the wave function is localized in the sense that is must approach zero as x increases toward infinity. This just means that the probability of finding the particle must decrease as we move toward very large extensions.

The solution to Schroedinger’s equation for the LHO is

An is a constant and Hn(q) is called a Hermite polynomial of the nth order

)()()( 22

2

xqqqq

)()( 2/2

qHeAq nq

nn 22

)1()( qn

nqn

n eq

eqH

Quantum Mechanical Linear Harmonic Oscillator

As with the Particle-in-a-Box, the probability of finding a particle at

By requiring that:

)()()( 22

2

xqqqq

)()( 2/2

qHeAq nq

nn 22

)1()( qn

nqn

n eq

eqH

qqH 2)(1 1)(0 qH 24)( 22 qqH q x

q x

2)(qP

1)( 2 dqqn

We find: 1/ 21

2 !n nA

n

Finally: 2

1/ 2

/ 2

2 !x

n nnx e H x

n

Quantum Mechanical Linear Harmonic Oscillator

The energy has the form:

Note this is shifted by h/2 from Planck’s energy. This is called the zero point energy, the existence of which is required by the Heisenberg Uncertainty Principle.

)()()( 22

2

xqqqq

2

1/ 2

/ 2

2 !x

n nnx e H x

n

124 n

hE

)21( nhvEn

Quantum Mechanical Linear Harmonic Oscillator

It is interesting to calculate probabilities Pn(x) for finding a harmonically oscillating particle with energy En at x; it is easier to work with the coordinate q; for n=0 we have:

2

1/ 2

/ 2

2 !x

n nnx e H x

n

2 2

2 2

2 2

2 2

1/ 22/ 2

0 0 0 0

1/ 2 22/ 2

1 1 1 1

221/ 222 / 2

2 2 2 2

231/ 223 / 2

3 3 3 3

1 1

2 2

2 11 2 12 2

2 31 2 33 3

q q

q q

q q

q q

q A e P q q e

qq A qe P q q e

qq A q e P q q e

q qq A q q e P q q e

Quantum Mechanical Linear Harmonic Oscillator

Probabilities: P(q) for n=0,1,2

00.10.20.30.40.50.6

q

P(x)

n=0n=1n=2

Quantum Mechanical Linear Harmonic Oscillator

Potential well, wave functions and probabilities:

E

2 2

28n hEmL

n = 1, 2, …

Wavefunctions for particle in the box

Quantum Mechanical Linear Harmonic Oscillator

Potential well, wave functions and probabilities:

As the quantum number gets larger, the probability increases towards larger displacement values. This corresponds to a classical phenomenon, as the energy of an oscillator increases, motion becomes more extended away from the status of lowest energy. The fundamental frequency of the oscillator is also the same both classically and in quantum mechanics:

Quantum Mechanical Linear Harmonic Oscillator

Potential well, wave functions and probabilities:

The fundamental frequency of the oscillator is also the same both classically and in quantum mechanics:

0

There are however several differences between classical and quantum mechanics

Quantum Mechanical Linear Harmonic Oscillator

Potential well, wave functions and probabilities:

the first is that there is motion even in the lowest energy state (see the shape of the probability for n=0); the second is that the wave function extend beyond the classical limits for the motion in a region of space where the potential is very large and that are not expected to be observed classically. The term 1/2hv is called zero point energy. This states that an oscillator cannot be at complete rest; if it was at rest, we would know the momentum (p=0) and position precisely; the zero point energy allows Heisenberg’s principle not to be violated. What that means is that molecules or solids vibrate even at 0 degree K.

Expectation values

The expectation value of a variable O can be calculated for a quantum system using the expression

We can calculate the expectation value for variables such as position and momentum; as in the classical oscillator

dxOO *

x pX 0

However, if we calculate the mean-square position and the mean-square momentum we will find some differences

Expectation values

In general

Thus, the squares are not zero although the expectation value for position and momentum are; in fact, they are related to the energy of the oscillator in the nth energy level.

Let us calculate the mean-square position and the mean-square momentum for a linear harmonic oscillator in the nth energy level as follows (notice that the wave functions are real and so complex conjugate is the same as the function itself)

2

1211)()(1 22 2220

22 dqeqdqeqdqqqqqx qq

2112 nx

21

2

22 nhp x