PH 333: Quantum Physics

Lecture 9

Matter Waves

Lecturer: Dr. Rebekah D’Arcy

Recap

420

222 cmcpE

E = h = hc/ʋ λλ = h/p

p = mv p = γm v

De Broglie’s Explanation of Quantization in the Bohr Model

The Bohr quantum condition for stationary states can be understood by requiring the circular path of the electrons around the nucleus to be an integral number of deBroglie wavelengths.

Any other “non-integral” path will cause total destructive interference.

Davisson and Germer

n = d Sin

What do de Broglie waves look like ?De Broglie gave the formula for and k, but he did not indicate what

they would look like.

Consider, a wave described by the wavefunction,(x,t), where.

If we plot the real part (or complex part):

It is difficult to see how this can represent a single particleThe wave is spread to with uniform amplitudeWaves of single and k are poor representatives for a single particle.

)(),( tkxietx

Representing a particle with matter waves:

(a) particle of mass m and speed v0

(b) superposition of many matter waves with a spread of wavelengths centered on λ0 = h/mv correctly represents a particle.

Consider a wave made up of two separate waves

txkktxkktx )(sin)(sin),( 0000

txktxktx cossin2),( 00

Consider the case k >>dk >>,

Then the wavelength of the envelope is much longer than the mean wavelength of the two waves.

The formula for the envelope is cos (k x - t) which indicates that it is a wave travelling in the + x direction, with group velocity

kg

The rapidly oscillating part of the pattern is travelling at phase velocity.

0

0

kvp

If the relationship between and k for the two waves satisfies de Broglie’s relationship

Then, we can get a relationship between k and ω as follows:

as 0>>, and k0>>k.

Likewise,

Subtracting 2 from 1,

2

420222

cm

ck

2

4202

0200

20

2

42022

02

0

22

cm

ckkk

cmckk

Eqn 1

2

4202

0200

20 22

cm

ckkk Eqn 2

2

0

0

200 44

ck

k

ckk

Since 0 and k0 are average values of frequency and wave vector, we can write,

where E and p are values of the energy and momentum for the particle. But

where v is the velocity of the particle, therefore

2

0

0

200 44

ck

k

ckk

00 kpE

vc

k

mvmc

kpE

2

0

0

2

0

0

particletheofvelocitythevck

k 2

0

0

This composite wave has a nice feature in that its envelope moves at the velocity of the particle.

However, it is still not very good in that it does not indicate the location of the particle it describes.

What we would like, is a wave of the following form where the wave travels at the particle’s velocity.

particletheofvelocitythevck

k 2

0

0

v

Spatially localised wave packetFirst we consider localising the wave packet.

– We will ignore the time dependence of the wave function for the moment.

– We will consider this later when we try to address the velocity of the wave.

Consider

We will consider adding a large number of waves– The k values stretch in a quasi-continuous fashion

about k0. – As we consider a large number of k values we can

replace the summation with an integral

ikxk k eAx )(

A(k)

k0

20)()()(

0

kkikx ekAwheredkekAx

k

dkeedkekAx ikxkkikx

00

20)()(

dkeeedkeeeex xkkikkxikxikikxkkxik

00

02

0002

00)(

dkeedkeex

ixkkkk

xikix

kkkkxik

0

22

0

02

00

02

00)(

We can write:

2222220

20

0

222

2

2,

bbabbabaabaixkkkk

thenixbkka

dkeeedkeex

ixixkk

xik

ixixkk

xik

22

00

22

00 2

0

2

0

22)(

dkeee

dkeeedkeeex

kkxxik

ixkkxxikixkkxxik

0

4

0

24

0

24

2'0

2

0

2

02

0

2

02

0)(

20'0

ixkk

Let

where

'0

0 '0

22'0 kkzwheredzedke

k

zkk

2ze

dzedze z

k

z 2

'0

2

We need to carry out the integration. Even though k0’ is a complex number, this does not present any problem.

If (k0’)2 1/, then for z less than k0, the function

is vanishingly small. Therefore we can add thevanishingly small part to the integral.

z = -k0’

z →

dze z2

Hence

4

0

42

02'

0

2

0)(xxikkkxxik eedkeeex

To summarise

We have succeeded in localising the wave packet in the vicinity of x = 0. This is suitable for a de Broglie wave representing a particle in the vicinity of x = 0.

x = 0Stationary wavepacket

4

2

0)(xxik eex

dkeex ikxkk

0

20)(

Uncertainty principle (space, momentum)

Before we consider developing a moving wavepacket, lets reflect a little more on the significance of what we have done.

To localise the wave we have had to add together many waves of different k values. In so doing we have introduced uncertainty in k or in momentum. Lets call the uncertainty in k, k.

From the above diagram we are also not sure where precisely the particle is localised. Let us consider this further.

x = 0Stationary wavepacket

2

422

0)(

xxik eex

244

22

0

2

0xxxikxxik eeeee

Consider the square of the wavefunction.

2/2 2

)( xecx Particle is localised to an

uncertainty here

x =0

x

Likewise, we can regard A(k) as indicating the range of k values which the particle is likely to have

The range of k values is around

k0

k0

1k

20 )(22 )( kkekA

Hence we have k x ≈ 1

*

Hence we have k x ≈ 1

The uncertainty in k and in position, x, is an inherent characteristic of describing matter using waves

Regarding matter waves, x is the uncertainty in the particle’s positionћk is p, the uncertainty in the particle’s momentum

p x ≈ ћ

Travelling wave packetNext consider the construction of a travelling wave packet.Consider

where k and are related by the deBroglie relationship

Lets assume that the width k of the wave packet is small so that there is only a narrow range of k values about k = k0 that are significant.

Likewise, assume only a narrow range of values around 0 is significant.

We can express in terms of k as follows:

dkeetx tkxikk

0

20),(

2

420222

cmck

.......)( 000

kkdkdkk

kk

000

)( kkdkdk

k

0kdkd

particleofvelocityvE

cpckdkd

kdkcd

cmck

k

,

22

00

20

0

20

2

2

420222

0

To find we proceed as follows:

p0 and E0 are the average values of the momentum and energy, respectively.

000)( kkvk

dkeetx tkxikk

0

20),(

dkeetx tkkvkxikk

0

0002

0),(

dkeeeetx tkkivtiikxkk

0

0002

0),(

dkeeeeetx tkkivtixikxkkikk

0

000002

0),(

dkeeeetx tkkivxkkikktxki

0

0002

000),(

dkeeetx tvxkkikktxki

0

002

000),(

To construct the travelling wave packet, we use the same approach as before. We consider,

dkeeetx tvxkkikktxki

0

002

000),(

tvxikkwheredkeeetx kktvx

txki00

''0

0

4

2),(

2''0

20

00

44

20

00

20

00),(tvx

txkitvx

txki eeeetx

v

Using the same approach as before, we can write down the sum as follows:

This is identical to previous wave packet except that the centre of the packet moves with velocity v0.

Do this!

Uncertainty principle, (energy, time)Because of the uncertainty in k there is uncertainty in as (k),

= k v0

And, because there is uncertainty in position, x, there is uncertainty in the time that the particle arrives at a specific point,

Hence

t = k x 1ћ t ћE t ћ

In Summary: We managed to construct wave functions which describe particles, but it is very tedious!

We also showed the uncertainty principle along the way!

Next day: We consider an alternative approach, is there a wave equation from which we can solve or extract solutions?

xv

t 0

1