superposition for the particle-in-a-box
TRANSCRIPT
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4.1 Time evolution ofsuperpositions
Slides: Video 4.1.2 Superposition forthe particle in a box
Text reference: Quantum Mechanicsfor Scientists and Engineers
Section 3.6 (“Simple linearsuperposition in an infinite
potential well”)
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Time evolution of superposi
Superposition for a particle in
Quantum mechanics for scientists and engineers Da
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Superposition for a particle in a box
Suppose we have an infinitely deep potentiawell
a “particle in a box”with the particle in a linear superpositi
for example, with equal parts of thefirst and second states of the well
1 21 , exp sin exp si
z z
E E z z t i t i t
L L
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Superposition for a particle in a box
From this superposition
we can multiply it by its complex conjugato get the probability density
1 2
1 , exp sin exp sin
z z
E E z
z t i t i t L L
2
2 2 2 1
,1 2
sin sin 2 cos sin z z z z
z t
E E z z z t
L L L L
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2 2 2 1 2 2
sin sin sin sin exp exp z z z z
z z z z E E i t
L L L L
Superposition for a particle in a box
1 2 2exp sin exp sin
z z
E z E zi t i t
L L
2 2 2 12 2
sin sin 2cos sin sin z z z z
z z E E z z
t L L L L
1 2 2exp sin exp sin
z z
E z E zi t i t
L L
multiplied bcomplex con
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Superposition for a particle in a box
Note this probability density
has a part that is oscillating in time
at an angular frequencyNote also that the absolute energy origin do
not matter here for this measurable quant
only the energy difference matter
2
2 2 2 1
,
1 2 sin sin 2 cos sin si
z z z z
z t
E E z z z t
L L L L
21 2 1 E E
2 1 E E
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Particle in a box
As a reminder
here are the first few particle-
in-a-box energy levelsand their associated
wavefunctionsplotted with the orange
dashed lines ashorizontal axes
E n e r g y
1n
2n
3n
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1n 1 z
Superposition
The spatial eigenfunction
is plotted here
with the bottom of the box
as its horizontal axis W a v e
f u n c t i o
n
z
1n
1 z
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Superposition
For the probability density
note the different shape
Multiplying 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 y
z
1n 1 z
11 1, exp
E z t i t z
2 21 1, z t z
1 , z
2
1 z
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2n 2 z
2n
2 z
Superposition
Similarly
The spatial eigenfunction
is plotted here
with the bottom of the boxas its horizontal axis W
a v e
f u n c t i o
n
z
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2n 2 z
2
2 z
Superposition
The probability density
is a positive function
Multiplying 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 y
z
22 2, exp
E z t i t z
2 22 2, z t z
2 , z
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Superposition
An equal superposition of thetwo oscillates
at the angular frequency
2 2
1 2
2 2
1 2
2 1
1 2
, , ,
2cos
z t z t z t
z z
E E t z z
21 2 1 1/ 3 / E E E
P
r o b a b i l i t y d e n s i t y
z
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P
r o b a b i l i t y d e n s i t y
Superposition
An equal superposition of thetwo oscillates
at the angular frequency
z
1 , z t
2 2
1 2
2 2
1 2
2 1
1 2
, , ,
2cos
z t z t z t
z z
E E t z z
21 2 1 1/ 3 / E E E
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