lesson 4 swe
TRANSCRIPT
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Quantum Mechanics for
Scientists and Engineers
David Miller
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Schrdingers equation
From de Broglie to Schrdinger
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Electrons as waves
de Broglies hypothesis is that the electron
wavelength is given by
wherep is the electron momentum andh is Plancks constant
J s
Now we want to use this to help construct a
wave equation
h
p
346.62606957 10h
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A Helmholtz wave equation
If we are considering only waves of one
wavelength for the momenti.e., monochromatic waves
we can choose a Helmholtz wave equation
with
which we know works for simple waves
with solutions like
sin(kz), cos(kz), and exp(ikz)(and sin(kz), cos(kz), and exp(ikz))
2
22d k
dz 2k
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A Helmholtz wave equation
In three dimensions, we can write this as
which has solutions likesin(k r), cos(k r), and exp(ik r)(and sin(-k r), cos(-k r), and exp(-ik r))
where k and r are vectors
2 2 22 2
2 2 2 k
x y z
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From Helmholtz to Schrdinger
With de Broglies hypothesis
and the definitionthen
where we have defined
soHence we can rewrite our Helmholtz equation
or
/h p
2 /k
2 2 2
/k p
2 / /k p h p
/ 2h
22
2
p
2 2 2p
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From Helmholtz to Schrdinger
If we are thinking of an electron, we can
divide both sides by its mass mo to obtain
But we know from classical mechanics that
and in general
2 22
2 2o o
p
m m
2
2kinetic energy of electron
o
p
m
Total energy ( )=Kinetic energy + Potential energy ( )E V r
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From Helmholtz to Schrdinger
So
Hence our Helmholtz equation
becomes the Schrdinger equation
or equivalently
2 22
2 2o o
p
m m
2 / 2Kinetic energy =
= Total energy ( ) - Potential energy ( )
op m
E V r
2
2
2o
E Vm
r
2
2
2o
V Em
r
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Schrdingers time-independent equation
We can postulate a Schrdinger equation for
any particle of mass m
Formally, this is the
time-independent Schrdinger equation
2
2
2
V E
m
r
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Probability densities
Borns postulate is that
the probability of finding an electronnear any specific point r in space
is proportional to the modulus squared
of the wave amplitudecan therefore be viewed as a
probability density
with called a probability amplitude
or a quantum mechanical amplitude
P r
2
r
r
2
r
r
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Schrdingers equation
Diffraction by two slits
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Youngs slits
An opaque mask has two slits cut in it, a distance s apart
s
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Youngs slits
We shine a plane wave on the mask from the left
s
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Youngs slits
What will be the pattern on a screen at a large distancezo?
s
oz
?
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Youngs slits
The slits as point sources give an interference pattern
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Youngs slits
The distance from the upper source to point x
on the screen is 2
2/ 2 ox s z
oz
2
s
x/ 2x s
2 22 2/ 2 1 / 2 /
o o ox s z z x s z
2/ 2 / 2o oz x s z 2 2/ 2 / 8 / 2
o o o oz x z s z sx z
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Youngs slits
The distance from the lower source to point x
on the screen is 2 2
/ 2 ox s z
oz
2
s
x
/ 2x s
2
/ 2 / 2o o
z x s z 2 2/ 2 / 8 / 2
o o o oz x z s z sx z
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Youngs slits
For large zo the waves are approximately uniformly bright
i.e., using exponential waves for convenience
Using our approximate formulas for the distances gives
where
2 22 2exp / 2 exp / 2
s o ox ik x s z ik x s z
exp exp / 2 exp / 2s o ox i ik sx z ik sx z 2 2/ 2 / 8o o ok z x z s z
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Youngs slits
Now
so
so the intensity of the beam
exp exp 2cosi i
exp exp exp2 2
s
o o
sx sxx i ik ik
z z
2
2 1
cos / 1 cos 2 /2
s o ox sx z sx z
exp cos exp cos2
o o
sx sxi k iz z
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Youngs slits
The interference fringes are spaced by /s od z s
sd
s
oz
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Youngs slits
This allows us to measure small wavelengths /s od s z
sd
s
oz
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Schrdingers equation
Interpreting diffraction by two slits
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Youngs slits
If the upper slit is blocked no interference pattern
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Youngs slits
If the lower slit is blocked no interference pattern
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