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3.1 Particles and barriers

Slides: Video 3.1.5 Barriers and

boundary conditionsText reference: Quantum Mechanics

for Scientists and EngineersSection 2.8

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Particles and barriers

Barriers and boundary condi

Quantum mechanics for scientists and engineers Da

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For our Schrödinger equation

if we presume that  E , V and   are finite

then must be finite also, so

If there was a jump in

then would be infinite at that po

Boundary conditions

22

22

d z

V z z E zm dz

 

 

2 2/d dz 

/d dz 2 2

/d dz 

/d dz  must be continuous

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Boundary conditions

Alsomust be finite

otherwise could be infinitebeing the limit of a difference

involving infinite quantitiesFor to be finite

/d dz 

2 2/d dz 

/d dz 

  must be continuous

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Boundary conditions

Now that we have these two boundaryconditions

we can proceed to solve problems with finit“heights” of boundaries

/d dz  must be continuous

  must be continuous

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Infinitely thick barrier

Suppose we have a barrier of heightV owith potential 0 to the left of the

barrier

A quantum mechanical wave isincident from the left

The energy  E of this wave ispositivei.e.,

        E      n      e      r      g      y

V

 z

0 E  

0

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Infinitely thick barrier

We allow for reflection from the barrierinto the region on the left

Using the general solution on the leftwith complex exponential waves

where, as beforeC exp(ikz) is the incident wave, going right

 Dexp(-ikz) is the reflected wave, going left

        E      n      e      r      g      y

V

 z

exp expleft 

  z C ikz D ikz   

2

2 /k mE   

  0

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Infinitely thick barrier

Presume thati.e., the incident wave energy is less

than the barrier heightInside the barrier, the wave equation

is

i.e., mathematically

        E      n      e      r      g      y

V

 z

o E V 

22

22  o

d zV z E z

m dz

   

2

2 2

2o

d z   m

V E zdz

 

 

0

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Infinitely thick barrier

The general solution of 

for the wave on the right is

whereWe presume

otherwise the wave increasesexponentially to the right for ever

        E      n      e      r      g      y

V

0

 z

2

2 22 o

d z   mV E z

dz

  

exp expright 

  z F z G z  

  2

2 /om V E     

0F  

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Infinitely thick barrier

Hence the wave on the rightinside the barrier, is

with

This solution proposes that the waveinside the barrier is not zeroInstead, it falls off exponentially

        E      n      e      r      g      y

V

0

 z

expright 

  z G z  

  22 /o

m V E     

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Infinitely thick barrier

Using the boundary conditionswe complete the solution

On the left, we haveOn the right we have

Continuity of the wavefunctionat gives

Continuity of the wavefunctionderivative at givesi.e.,

exp left    z C ikz     exp

right   z G z  

0 z    C D G

0 z i

C D Gk 

 

ikC ikD G 

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Infinitely thick barrier

Adding

gives

Equivalently

so we have found the amplitude G of the wave in t

in terms of the amplitude C of the incident wave

C D G i

C D Gk 

 

2 1  i

C G k

22  k 

G C k i

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Infinitely thick barrier

Subtracting

gives

Since we found

then

so we have found the amplitude  D of the reflected

terms of the amplitude C of the incident wave

C D G i

C D Gk 

 

2 1  i

 D G

k

2 /G k k i C    k i

 D C k i

 

 

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Infinitely thick barrier

We have now formally solvedthe problem

the wave on the left iswith

the wave on the right is

with

where

exp left    z C ikz     k i

 D C 

k i

 

 

exp

right   z G z  

2k 

G C k i 

2 m   22 /k mE   

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Reflection at an infinitely thick barrier

Note that

so

so

so the barrier is 100% reflectingthough there is a phase shift on r

an effect with no classical analo

k i D C 

k i

 

 

 D k i

C k i

 

 

2

1 D k i k i

C k i k i

 

 

f

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Wavefunction at a barrier

We plot the real part of thewavefunction

for an electronenergy 1 eV

at a barrierheight 2 eV

Notethe wavefunctionand its derivative

are both continuous

2   0

z (nm)

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W f i b i

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Wavefunction at a barrier

Note the exponential decaylength

2   0

z (nm)

0.2 nm

1/ 0.2 2nm Å  

W f ti t b i

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Wavefunction at a barrier

As we in

energythe expdecay

P b bilit d it t b i

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2   0

z (nm)

Probability density at a barrier

Probability densityWith

then

falling by 1/e in1/ 2 0.1 1nm Å   

expright 

  z z  

2

exp 2right   z z  

 2

right   z 

0.1 nm

P b bilit d it t b i

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Probability density at a barrier

As we inenergythe exp

decay

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