Physics 361Principles of Modern Physics

Lecture 14

Solving Problems with the Schrödinger Equation

This lecture• Harmonic oscillatorNext lecture• Tunneling and scattering in 1D

Finite depth box – First consider an infinite box potential

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These energy levels increase in separation as n increases.

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Finite depth box – What happens to the energy levels?

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Remember what happens to wave function in potential barrier as energy increases

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A single potential barrier as energy increasesThis gives

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oStanding wave pattern Exponential decay

As energy increases

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oStanding wave patternfrequency increases.

Exponential decay lengthincrease.

Physical argument for this is that as the particle becomes more energetic, it penetrates into the barrier further.

Finite depth box – What happens to the energy levels?

When the particles can tunnel into the finite barrier, the effective width of the well is greaterthan .

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For an infinite box potential, the walls of the potential set the size of the oscillatory waves

Finite depth box – energy levels are decreased

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oFor an infinite box potential the wave functions are less constrained by the potential barrier as the energy approaches the potential barrier height – the particle tunnels into the barrier more.

Starting with the energy infinite box states

we can approximate the effective well widthby adding the tunneling depth with a constantof order unity.

Taylor expansion for small changes gives a lowering of the energy levels. This lowering is more pronounced for the higher-energy states.

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Harmonic oscillator potential – this is an extremely important potential to study.

Consider any arbitrary potential landscape

Where would particles end up if you threw a bunch at this potential landscape?

Harmonic oscillator potential – this is an extremely important potential to study.

Consider any arbitrary potential landscape

Where would particles end up if you threw a bunch at this potential landscape?

At the minimum in thermal equilibrium!!

Thus, the behaviors near the minima are physically very important.

Consider potential near a single minimum.

What is the general behavior near any local minimum?

Consider potential near a single minimum.

What is the general behavior near any local minimum?

To see general behavior look at Taylor expansion about minimum.

What can we say about the terms?

Consider potential near a single minimum.

What is the general behavior near any local minimum?

The slope is zero at the minimum, so first derivative is zero there as well. Thus, we can approximate

as

since

Harmonic oscillator potential

We can approximate the potential as

which is the same as the harmonic oscillator potential

This is usually written in terms of the oscillation frequency of the oscillatorwhich gives

Harmonic oscillator potential energy levels

Classical turning points are intersections of energy with potential

Use this classical width as an estimate of the width of the box for the energy levels. Go back to infinite box energy states.

Now the classical turning points for a harmonic oscillator are given by

Solving this relation for the length we have

Now we just insert this length in for the length in the box energy level relation.

Harmonic oscillator potential energy levels

Inserting the classical turning points width, we obtain

which gives approximately

These are energy levels which are equally spaced!!

The full exact solution gives the same with a lowerminimum energy of .