superposition of the harmonic oscillator
TRANSCRIPT
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4.1 Time evolution ofsuperpositions
Slides: Video 4.1.4 Superposition for
the harmonic oscillatorText reference: Quantum Mechanics
for Scientists and EngineersSection 3.6 (“Harmonic oscillator
example”)
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Time evolution of superposition
Superposition for the harmonic osci
Quantum mechanics for scientists and engineers David M
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Superpositions and oscillation
Quite generallyif we make a linear combination of two
energy eigenstateswith energies E a and E b
the resulting probability distributionwill oscillate at the (angular)frequency
/ab a b E E
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Superpositions and oscillation
So, if we have a superposition wavefunction
then the probability distribution will be
where
, exp expa b
ab a a b b
E E
t c i t c i t
r r
2 2 22 2,
2 cos
ab a a b b
a b
a a b b
t c c
E E t c c
r r r
r r
argab a a b bc c r r
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Harmonic oscillator
As a reminderhere are the first two
harmonic energy levelsand their associated
wavefunctionsplotted with the orange
dashed lines ashorizontal axes
E n e r g y
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Superposition
The spatial eigenfunction
is plotted herewith the bottom of the
parabolic well as itshorizontal axis W
a v e f u n c t i o n
0n 0n 0 z
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Superposition
For the probability density
note the narrower shapeMultiplying by the time
dependent factor gives
The probability densities arethe same
P
r o b
a b i l i t y d
e n s
i t y0n
00 0, exp
E z t i t z
2 20 0, z t z
2
0 z
0 z
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Superposition
The spatial eigenfunction
is plotted herewith the bottom of the
parabolic well as itshorizontal axis W
a v e f u n c t i o n
1n 1n 1 z
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Superposition
For the probability density
note it is positiveMultiplying by the time
dependent factor gives
The probability densities arethe same
1n
P r o b a b
i l i t y d e n s i t y
11 1, exp
E z t i t z
2 21 1, z t z
2
1 z
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Superposition
An equal superposition of thetwo oscillatesat the angular frequency
2 2
0 1
2 2
0 1
0 1
, , ,
2cos
z t z t z t
z z
t z z
1 0 / E E
P r o b a b
i l i t y d e n s i t y
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Superposition
An equal superposition of thetwo oscillatesat the angular frequency
0 1, z t
2 2
0 1
2 2
0 1
0 1
, , ,
2cos
z t z t z t
z z
t z z
1 0 / E E
P r o b a b
i l i t y d e n s i t y
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