Post on 10-Apr-2016
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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