LECTURE 13

MATTER WAVES

WAVEPACKETS

PHYS 420-SPRING 2006

Dennis Papadopoulos

Fig. 4-22, p. 132

Fig. 4-21, p. 131

Fig. 4-15, p. 127

Fig. 4-20, p. 129

)2( 22

2

n

nC

Balmer Series

Fig. 4-23, p. 133

Fig. 4-24, p. 134

We said earlier that Bohr was mostly right…so where did he go

wrong?

• Failed to account for why some spectral lines are stronger than others. (To determine transition probabilities, you need QUANTUM MECHANICS!) Auugh!

• Treats an electron like a miniature planet…but is an electron a particle…or a wave?

2/

...3,2,1

h

nnvrme

In order to understand quantum mechanics, you must understand waves!

an integer number of wavelengths fits into the circular orbit

rn 2

where

p

h

is the de Broglie wavelength

Photonsp=h/c=

h/

Phase cancellation

“Okay…what you have to realize is…to first order…everything is a simple harmonic oscillator. Once you’ve got that, it’s all downhill from there.”

An oscillation is a time-varying disturbance.

oscillation

kxF :forcerestoring

wave

A wave is a time-varying disturbance that also propagates in space.

tkxAy

tAy

sin

becomessin

(but they are nonetheless instructive)

fv

ftx

Ay

p

2

2cos

/2,2

cos

kf

tkxAy

A wave that propagates forever in one dimension is described by:

in shorthand:

angular frequency

wave number

waves can interfere (add or cancel)

Standing Waves

“Beats” occur when you add two waves of slightly different

frequency. They will interfere constructively

in some areas and destructively in others.

txkktxkkAy

txkAtxkAyyy

21211212

221121

2

1cos(

2

1cos2

coscos

Interefering waves, generally…

Can be interpreted as a sinusoidal envelope:

txk

A22

cos2

Modulating a high frequency wave within the envelope:

txkk 2121 2

1

2

1cos

kkkg

2/

2/

12

12the group velocity

1

1

1

12

12

2/

2/

kkkpthe phase

velocity

if 21

kkkg

2/

2/

12

12the group velocity

Beats

http://www.school-for-champions.com/science/soundbeat.htm

http://library.thinkquest.org/19537/java/Beats.html

Standing waves (harmonics)

Ends (or edges) must stay fixed. That’s what we call a boundary condition.

This is an example of a Bessel function.

Legendre’s equation:

comes up in solving the hydrogen atom

It has solutions of:

de Broglie’s concept of an atom…

Bessel Functions:

are simply the solution to Bessel’s equations:

Occurs in problems with cylindrical symmetry involving electric fields, vibrations, heat conduction, optical diffraction.

Spherical Bessel functions arise in problems with spherical symmetry.

The word “particle” in the phrase “wave-particle duality” suggests that this wave is somewhat localized.

How do we describe this mathematically?

…or this

…or this

FOURIER THEOREM: any wave packet can be expressed as a superposition of an infinite

number of harmonic waves

FOURIER THEOREM: any wave packet can be expressed as a superposition of an infinite

number of harmonic waves

dkekaxf ikx)(

2

1)(

spatially localized wave group

amplitude of wave with wavenumber k=2/

adding varying amounts of an infinite number of

wavessinusoidal expression

for harmonics

Adding several waves of different wavelengths

together will produce an interference pattern which

begins to localize the wave.

To form a pulse that is zero everywhere outside of a finite spatial range x requires adding together

an infinite number of waves with continuously varying wavelengths and

amplitudes.

Remember our sine wave that went on “forever”?

We knew its momentum very precisely, because the momentum is a function of the frequency, and the frequency was very well defined.

But what is the frequency of our localized wave packet? We had to add a bunch of waves of different frequencies to produce it.

Consequence: The more localized the wave packet, the less precisely defined the momentum.

Consequence: The more localized the wave packet, the less precisely defined the momentum.

kp

E

How does this wave behave at a boundary?

at a free (soft) boundary, the restoring force is zero and the reflected wave has the same polarity (no phase change) as the incident wave

at a fixed (hard) boundary, the displacement remains zero and the reflected wave changes its polarity (undergoes a 180o phase change)

When a wave encounters a boundary which is neither rigid (hard) nor free (soft) but instead somewhere in between, part of the wave is reflected from the boundary and part of the wave is transmitted across the boundary

In this animation, the density of the thick string is four times that of the thin string …