Particle in a Box

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Particle in a Box
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Class Objectives

• Introduce the idea of a free particle.
• Solve the TISE for the free particle case.

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Particle in a Box

• The best way to understand Schrödingers equation
  is to solve it for various potentials.

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Particle in a Box

• The best way to understand Schrödingers equation
  is to solve it for various potentials.
• The simplest of these involving forces is
  particle confinement (a particle in a box).

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Particle in a Box

• Consider a particle confined along the x axis
  between the points x 0 and x L.

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Particle in a Box

• Consider a particle confined along the x axis
  between the points x 0 and x L.
• Inside the box it free but at the edges it
  experiences strong forces which keep it confined.

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Particle in a Box

• Consider a particle confined along the x axis
  between the points x 0 and x L.
• Inside the box it free but at the edges it
  experiences strong forces which keep it confined.
  Eg. A ball bouncing between 2 impenetrable walls.

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Particle in a Box

• Consider a particle confined along the x axis
  between the points x 0 and x L.
• Inside the box it free but at the edges it
  experiences strong forces which keep it confined.
  Eg. A ball bouncing between 2 impenetrable walls.

U
E
0
L
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Particle in a Box
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Particle in a Box

• Inside the well the particle is free.

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Particle in a Box

• Inside the well the particle is free.
• This is because is zero inside the well.

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Particle in a Box

• Inside the well the particle is free.
• This is because is zero inside the well.
• Increasing to infinity as the width is
  reduced to zero, we have the idealization of an
  infinite potential square well.

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Infinite square potential
U
L
0
x
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Particle in a Box

• Classically there is no restriction on the energy
  or momentum of the particle.

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Particle in a Box

• Classically there is no restriction on the energy
  or momentum of the particle.
• However from QM we have energy quantization.

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Particle in a Box

• We are interested in the time independent
  waveform of the particle.
• The particle can never be found outside the well.
  Ie. in the region

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Particle in a Box

• Since we get that,

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Particle in a Box

• Since we get that,
• So that

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Particle in a Box

• The solutions to this equation are of the form
  for
  (a linear combination of cosine and sine waves of
  wave number k)

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Particle in a Box

• The solutions to this equation are of the form
  for
  (a linear combination of cosine and sine waves of
  wave number k)
• The interior wave must match the exterior wave at
  the boundaries of the well. Ie. To be continuous!

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Particle in a Box

• Therefore the wave must be zero at the
  boundaries, x0 and xL.

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Particle in a Box

• Therefore the wave must be zero at the
  boundaries, x0 and xL.
• At x0,

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Particle in a Box

• Therefore the wave must be zero at the
  boundaries, x0 and xL.
• At x0,

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Particle in a Box

• Therefore the wave must be zero at the
  boundaries, x0 and xL.
• At x0,

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Particle in a Box

• Therefore the wave must be zero at the
  boundaries, x0 and xL.
• At x0,
• At xL,

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Particle in a Box

• Therefore the wave must be zero at the
  boundaries, x0 and xL.
• At x0,
• At xL,
• Since , then
• ,
• Recall

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Particle in a Box

• From this we find that particle energy is
  quantized. The restricted values are

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Particle in a Box

• From this we find that particle energy is
  quantized. The restricted values are
• Note E0 is not allowed!

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Particle in a Box

• From this we find that particle energy is
  quantized. The restricted values are
• Note E0 is not allowed!
• N1 is ground state and n2,3 excited states.

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Particle in a Box

• Finally, given k and B we write the waveform as

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Particle in a Box

• Finally, given k and B we write the waveform as
• We need to determine A.

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Particle in a Box

• Finally, given k and B we write the waveform as
• We need to determine A.
• To do this we need to normalise.

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Particle in a Box

• Normalising,

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Particle in a Box

• Normalising,

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Particle in a Box

• Normalising,

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Particle in a Box

• Normalising,

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Particle in a Box

• Normalising,

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Particle in a Box

• Normalising,

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Particle in a Box

• For each value of the quantum number n there is a
  specific waveform describing the state of a
  particle with energy .

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Particle in a Box

• For each value of the quantum number n there is a
  specific waveform describing the state of a
  particle with energy .
• The following are plots of vs x and the
  probability density vs x.

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Particle in a Box