Barriers and Boundary Conditions

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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



Particles and barriers

Barriers and boundary condi

Quantum mechanics for scientists and engineers Da



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



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



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



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




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




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




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




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




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




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




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




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



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



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)



W f i b i




Note the exponential decaylength

2 0

z (nm)

0.2 nm

1/ 0.2 2nm Å

W f ti t b i




As we in

energythe expdecay

P b bilit d it t b i



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



Probability density at a barrier

As we inenergythe exp

decay



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Barriers and Boundary Conditions

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