2.3.4_slides solving for the particle-in-a-box

8/10/2019 2.3.4_slides Solving for the Particle-In-A-box

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2.3 Particle in a box

Slides: Video 2.3.4 Solving for the

particle in a boxText reference: Quantum Mechanics

for Scientists and EngineersSection 2.6 (first part)



The particle in a box

Solving for the particle in a bo

Quantum mechanics for scientists and engineers David M



Particle in a box

We consider a particle of mass m

with a spatially-varying potential V ( z ) in th z direction

so we have a Schrödinger equation

where E is the energy of the particleand ( z ) is the wavefunction

22

22

d zV z z E z

m dz



Particle in a box

Suppose the potential energy is asimple “rectangular” potential well

thickness L z

Potential energy is constant inside

we choose thererising to infinity at the walls

i.e., at andWe will sometimes call thisan infinite or infinitely deep

(potential) well

E n e r g y0V

0 z z z L

0 z

0V



Particle in a box

Because these potentials at andat are infinitely highbut the particle’s energy E is

presumably finite

we presume there is no possibilityof finding the particle outside

i.e., for orso the wavefunction is 0 thereso should be 0 at the walls

E n e r g y

0 z z z L

0 z

0 z

z z L

0V



Particle in a box

With these choicesinside the well

the Schrödinger equation

becomes

with the boundary conditions

and

E n e r g y

0 z

0V

22

22

d zV z z E z

m dz

22

22

d z E z

m dz

0 0 0 z L



Particle in a box

The general solution to the equation

is of the form

where A and B are constants

andThe boundary condition

means because

E n e r g y

0 z

0V

22

22

d z E zm dz

sin cos z A kz B kz

22 /k mE

0 0

0 B cos 0 1



Particle in a box

With nowand the condition

kz must be a multiple of , i.e.,

where n is an integerSince, therefore,

the solutions are

with

E n e r g y

sin z A kz 0 z L

22 / / zk mE n L

2 2

2

k E

m

sinn n z

n z

z A L

22

2n z

n

E m L

1n

2n

3n



Particle in a box

We restrict n to positive integersfor the following reasonsSince for any real

number a

the wavefunctions with negative nare the same as those withpositive n

within an arbitrary factor, here -1the wavefunction for is trivial

the wavefunction is 0 everywhere

E n e r g y

1n

2n

3n

1, 2,n

sin sina a

0n

n z A



Particle in a box

We can normalize the wavefunctions

To have this integral equal 1choose

Note A n can be complex

All such solutions are arbitrarywithin a unit complex factor

Conventionally, we choose A n

real for simplicity in writing

E n e r g y

1n

2n

3n 2 22

0sin 2

z L

zn n

z

Ln z A dz A L

2 /n z L



Particle in a box

E n e r g y

1n

2n

3n

2sinn

z z

n z z

L L

22

2n

z

n E m L

1, 2,n 0

2.3.4_slides solving for the particle-in-a-box

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