wavepacket1 reading: qm course packet free particle gaussian wavepacket
TRANSCRIPT
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Wavepacket 1
Reading:QM Course packet
FREE PARTICLE GAUSSIAN WAVEPACKET
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Wavepacket 2
• Time dependent Schrödinger equation• Energy eigenvalue equation (time independent SE)• Eigenstates • Time dependence• (Connection to separation of variables)• Mathematical representations of the above
GAUSSIAN WAVE PACKET - REVIEW
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Wavepacket 3
Build a "wavepacket" from free particle eigenstates
•Ask 2 important questions:•Given a particular superposition, what can we learn about the particle's location and momentum?
HEISENBERG UNCERTAINTY PRINCIPLE
•How does the wavepacket evolve in time?
GROUP VELOCITY
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Wavepacket 4
We have already discussed the principle of superposition & the time evolution of that superposition in the context of the discrete quantum mechanical states of the infinite potential energy well.
We have also already discussed how build a Gaussian wave packet from harmonic waveforms (with a continuous frequency distribution) in the context of the classical rope problem.
We now look at the case of superposition of quantum mechanical states of the free particle, which are no longer discrete.
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Wavepacket 5
Free particle eigenstates
•Oscillating function•Definite momentum p = hk/2π•Subscript k on reminds us that depends on k
• What are E and in terms of given quantities?
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Wavepacket 6
Superposition of eigenstates (Fourier series)
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Wavepacket 7
Superposition of eigenstates(Fourier integral)
Localized particleIndefinite momentum
Definite momentumExtended position
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Wavepacket 8
Superposition of eigenstatesHow does it develop in time?
Definite momentumExtended position
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Localized particleIndefinite momentum
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Wavepacket 9
Gaussian wave packet
Localized particle
What is A(k)? x
We'll ignore overall constants that are not of primary importance (there are conventions about factors of 2π that are important to take care of to get numerical results, but we're after the physics!)
Projection of general function on eigenstate
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Wavepacket 10
Gaussian wave packet
Localized particle
x
We did this integral (by hand).
using:
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Wavepacket 11
Gaussian wave packet
Localized particle
x
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Wavepacket 12
Gaussian wave packet
Localized particle
x
k
If (x) is wide, A(k) is narrow and vice versa.
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Wavepacket 13
A(k) is the projection of (x) on the momentum eigenstates eikx, and thus represents the amplitude of each momentum eigenstate in the superposition.We need the contribution of a wide spread of momentum states to localize a particle. If we have the contribution of just a few, the location of the particle is uncertain
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Wavepacket 14
x
k
To define "uncertainty" in position or momentum, we must consider probability, not wave function.
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Wavepacket 15
x
k
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Wavepacket 16
HEISENBERG UNCERTAINTY PRINCIPLE
x
k
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Wavepacket 17
Next, we’ll ask how a general wave packet propagates,And deal with the particular example of the Gaussian wavepacket.
In short, we simply attach the exp(-iE(k)t/hbar) factor to each eigenstate and let time run.
Difference to non-dispersive equation: not all waves propagate with same velocity. “Packet” does not stay intact! Need to invoke “group velocity” to follow the progress of the “bump”.
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Wavepacket 18
Superposition of eigenstatesHow does it develop in time?
Localized particle
If depends on k, the different eigenstates (waves) making up this "packet" travel at different speeds, so the feature at x=0 that exists at t=0 may not stay intact at all time.It may stay identifiably intact for some reasonable time, and if it does, how fast does it travel?The answer is "it travels at the group velocity d/dk"
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Wavepacket 19
Superposition of eigenstatesHow does it develop in time?
Localized particle
The quantity d/dk may (does!) vary depending on the k value at which you choose to evaluate it. So it must be evaluated at a particular value k0 that represents the center of the packet. The next few pages spend time deriving the basic result. The derivation is not so important. The result is important:
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Wavepacket 20
For those of you who want more:
Localized particle
is a smooth function of kand k0
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Wavepacket 21
Localized particle
For those of you who want more:
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Wavepacket 22
Localized particle
Phase factor - goes to 1 in probability
For those of you who want more:
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Wavepacket 23
Localized particle
For those of you who want more:
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Wavepacket 24
Particular example of the Gaussian wavepacket.
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Wavepacket 25
Use same integral as before
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Wavepacket 26
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Wavepacket 27
Zero-momentum wavepacket:Spreads but doesn’t travel!It has many positive k components as it has negative k components.
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Wavepacket 28
is characteristic time for wavepacket to spread
for macroscopic things: m ≈ 10-3kg; 1/ ≈ 10-2 m ≈ 1027 s ≈ 1020 yr !! long
for nuclear scale: m ≈ 10-?kg; 1/ ≈ 10-? m ≈ 10-? s
for atomic scale: m ≈ 10-?kg; 1/ ≈ 10-? m ≈ 10-? s
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Wavepacket 29
• Gaussian superposition of free-particle eigenstates (of energy and momentum!)
• Localized in space means dispersed in momentum and vice versa.
• Look at time-dependent probability distribution: packet broadens and moves
FREE PARTICLE QUANTUM WAVEPACKET - REVIEW