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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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