Transcript lecture 19 (zipped power point) (update: 13Jan 04)

Recap: statistical interpretation of
radiation

The probability of observing a photon is
proportional to N (the number of photon
crossing a unit cross section in a unit time)
 Also, correspondence principle says
I  Nh   0c E 2

Hence the probability of observing a photon is
2
Prob (x)  E
Square of the mean of the square of
the wave field amplitude
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What is the physical interpretation
of matter wave?

we will call the mathematical representation of the de Broglie’s wave /
matter wave associated with a given particle (or an physical entity) as
The wave function, Y(x,t)


We wish to answer the following questions:
Where is exactly the particle located within Dx? the locality of a particle
becomes fuzzy when it’s represented by its matter wave. We can no
more tell for sure where it is exactly located.
 Recall that in the case of conventional wave physics, |field amplitude|2 is
proportional to the intensity of the wave). Now, what does |Y |2
physically mean?
2
Probabilistic interpretation of (the
square of) matter wave





As seen in the case of radiation field,
|electric field’s amplitude|2 is proportional to the
probability of finding a photon
In exact analogy to the statistical interpretation
of the radiation field,
P(x) = |Y |2 is interpreted as the probability
density of observing a material particle
More quantitatively,
Probability for a particle to be found between
point a and b is
b
b
a
a
p(a  x  b)   P( x)dx   | Y( x, t ) |2 dx
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Hence, a particle’s
wave function gives
rise to a probabilistic
interpretation of the
position of a particle
 Max Born in 1926

German-British physicist who worked on the mathematical
basis for quantum mechanics. Born's most important
contribution was his suggestion that the absolute square of
the wavefunction in the Schrödinger equation was a measure
of the probability of finding the particle at a given location.
Born shared the 1954 Nobel Prize in physics with Bothe
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Some weird philosophical
inferences of the probabilistic
interpretation

Due to the probabilistic interpretation of the
matter wave, the notion of “existence” of a
physical entity, at its most fundamental level,
begins to deviate from our conventional wisdom
 The existence of an entity is now no more be
deterministic notion (e.g. it either exist or not at
all) but only a “probability”
 If interested, please read the philosophical
interpretation of quantum mechanics yourself
 Its one of the most intriguing argument of the
last century and is still continue to be so
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Quantum description of a
particle in an infinite well


Imagine that we put particle
(e.g. an electron) into an
“infinite well” with width L (e.g.
a potential trap with sufficiently
high barrier)
In other words, the particle is
confined within 0 < x < L
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Another experimental scenario to trap a particle: using
electric potential trap: As V  infinity, the potential trap
approaches that of an idealised infinite quantum well
The charged particle moves freely inside the region of zero potential,
0 < x < L. But it would be bounced back when it bangs on the infinitely
“hard” wall. Mathematically this is described by saying that the potential
takes the form
, x  0, x  L

V ( x)  
 0,
0 x L
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Particle forms standing wave
within the infinite well
 How
would the wave
function of the particle
behave inside the well?
 They
form standing
waves which are
confined within
0≤x≤L
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Standing wave in general

Description of standing waves which ends are
fixed at x = 0 and x = L. (for standing wave, the
speed is constant), v = l = constant)
L = l1/2
(n = 1)
L
L = l2
(n = 2)
L = 3l3/2
(n = 3)
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