CH 6 SECTION 4
Particle-Wave Propagation
WAVE PULSE
• So far we’ve modeled a free particle as a single plane
wave:
Ψ(𝑥𝑥, 𝑡𝑡) = 𝐴𝐴𝑒𝑒 𝑖𝑖(𝑘𝑘𝑘𝑘−𝜔𝜔𝑡𝑡)
• This is NOT a realistic model for a single particle.
• A wave pulse is more realistic.
• A wave pulse is mathematically the sum of plane
waves.
MOTION OF A WAVE PULSE
•
A traveling particle is modeled as traveling wave
pulse.
•
A traveling wave pulse is the sum of plane waves,
refered to as a wave group.
•
The group velocity (dark red arrow) is the speed
of the particle.
•
The phase velocity (light arrow) is the speed of
the individual crests. NO INFORMATION IS
CARRIED AT THE PHASE VELOCITY SO IT IS
UNIMPORTANT.
•
We’ll get back to this Gaussian pulse, but first a
mathematically simpler wave group.
•
•
A SIMPLE WAVE GROUP
The simplest wave group is the sum of two plane waves:
Ψ 𝑥𝑥, 𝑡𝑡 = 𝐴𝐴𝑒𝑒 𝑖𝑖
𝑘𝑘1𝑥𝑥−𝜔𝜔1𝑡𝑡
+ 𝐴𝐴𝑒𝑒 𝑖𝑖(𝑘𝑘2𝑥𝑥−𝜔𝜔2𝑡𝑡)
where k and ω are just below and just above the central values:
k1 = k0 + dk , and k2 = k0 − dk
ω1 = ω0 + dω , and ω2 = ω0 − dω
•
Sub these into wave function and apply Euler’s equation:
Ψ 𝑥𝑥, 𝑡𝑡 = 2𝐴𝐴𝑒𝑒 𝑖𝑖
𝑘𝑘0𝑥𝑥−𝜔𝜔0𝑡𝑡
cos [ 𝑑𝑑𝑑𝑑 𝑥𝑥 − 𝑑𝑑𝑑𝑑 𝑡𝑡]
(This is the plot on the right)
•
•
𝑣𝑣phase =
𝜔𝜔0
𝑘𝑘0
and 𝑣𝑣group =
𝑑𝑑𝜔𝜔
𝑑𝑑𝑘𝑘
Probability density: does NOT depend on phase velocity
∗
Ψ Ψ = 4𝐴𝐴2cos2[ 𝑑𝑑𝑑𝑑 𝑥𝑥 − 𝑑𝑑𝑑𝑑 𝑡𝑡]
NOT A GOOD MODEL OF
A PARTICLE
A BETTER MODEL FOR A PARTICLE
•
In general a wave group is sum (integral) of plane waves:
•
From Chapter 4, 𝜓𝜓 𝑥𝑥 = ∫−∞ 𝐴𝐴(𝑘𝑘)𝑒𝑒 𝑖𝑖𝑖𝑖𝑖𝑖 𝑑𝑑𝑑𝑑
•
∞
Now extend this to include time dependence:
∞
Ψ 𝑥𝑥, 𝑡𝑡 = � 𝐴𝐴(𝑘𝑘)𝑒𝑒 𝑖𝑖(𝑘𝑘𝑘𝑘−𝜔𝜔𝜔𝜔) 𝑑𝑑𝑑𝑑
−∞
•
This wave packet (in the figure) happens to be a Gaussian.
•
At t = 0, we have the familiar:
2
•
Ψ 𝑥𝑥, 0 = 𝐶𝐶𝑒𝑒 −(𝑥𝑥/2𝜀𝜀) 𝑒𝑒 𝑖𝑖𝑘𝑘0 𝑥𝑥
The full time dependence function comes from integrating and it
is pretty messy. Let’s look at the probability density.
A BETTER MODEL FOR A PARTICLE
•
Probability density:
Ψ
•
2
=
− 𝑥𝑥 − 𝑠𝑠𝑠𝑠 2
exp
𝐷𝐷2𝑡𝑡2
1 + 𝐷𝐷2𝑡𝑡2/4𝜀𝜀4
2𝜀𝜀2(1 +
)
4𝜀𝜀4
𝐶𝐶2
Where, the group velocity and dispersion are:
𝑣𝑣group = 𝑠𝑠 ≡
𝑑𝑑𝑑𝑑
|
𝑑𝑑𝑑𝑑 𝑘𝑘0
•
More a dispersion in a moment.
•
The figure is for D = 0
and 𝐷𝐷 ≡
𝑑𝑑2𝜔𝜔
|
𝑑𝑑𝑑𝑑2 𝑘𝑘0
DISPERSION RELATIONS
•
A dispersion relation is an equation that expresses angular frequency ω in terms of
angular wave number k.
•
For electromagnetic waves, this expression is familiar: ω = ck ( you used it in many
problems when you wrote f = c/λ).
•
For a free particle, the plane wave solution has a familiar dispersion relation:
ℏ𝑘𝑘2
𝜔𝜔 =
2𝑚𝑚
PHASE AND GROUP VELOCITIES
•
Use dispersion relations and
definitions of phase and group
velocities.
•
𝑣𝑣phase =
𝜔𝜔0
𝑘𝑘0
and 𝑣𝑣group =
•
FOR ΕΜ : ω = ck
•
For matter: 𝜔𝜔 =
ℏ𝑘𝑘2
2𝑚𝑚
𝑑𝑑𝜔𝜔
𝑑𝑑𝑘𝑘
DISPERSION
•
•
•
•
Ψ
2
=
𝐶𝐶2
1+𝐷𝐷2𝑡𝑡2/4𝜀𝜀4
exp
− 𝑥𝑥−𝑠𝑠𝑠𝑠
2𝜀𝜀2(1+
2
2 2
𝐷𝐷 𝑡𝑡
4 )
4𝜀𝜀
D is the dispersion coefficient, 𝐷𝐷 ≡
𝑑𝑑2𝜔𝜔
|
𝑑𝑑𝑑𝑑2 𝑘𝑘0
If D is not zero (dispersion relation is
nonlinear), then the wave pulse spreads.
As it does so, the uncertainty particle’s
position increases.
IN CLASS PROBLEMS
•
CH 6: 47 AND 49
•
PLEASE WRITE ON THE BOARD.
AFTER I’VE REVIEWED YOUR
UNDERSTANDING, YOU CAN LEAVE.