Lecture 5

LECTURE 5

Source: D. Griffiths, Introduction to Quantum Mechanics (Prentice Hall, 2004)R. Scherrer, Quantum Mechanics An Accessible Introduction (Pearson Int’l Ed., 2006)R. Eisberg & R. Resnick, Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (Wiley, 1974)

Topics Today

• Bound States of Square Well E > 0.• Bound States of Square Well E < 0.

•Consider the potential

•This is a "square" well potential of width 2a and depth V0.

E > 0

Bound States of the Square Well

-Vo

axfor,0

axafor,V)x(V o

Bound States of the Square WellRegion I and III:

xΨEdxΨd

m2 2

22

0222

2

xEmdxd

Region II:

xΨExΨVdxΨd

m2 o2

22

0xΨEVm2dxΨd

o22

2

22 2

mEk

2o2 EVm2l

-Vo

E > 0

The solutions are, in general:

•Region I:                                                

•Region II:                                                         

•Region III:                                              In these the A's, B's, C’s and D's are constants.

Bound States of the Square Well, E > 0

Allowed energy for infinite square well

•Consider the potential

•This is a "square" well potential of width 2a and depth V0.

E < 0

Bound States of the Square Well

22 mE2

κ

-Vo

axfor,0

axafor,V)x(V o

Bound States of the Square WellRegion I and III:

xEdxd

m

2

22

2

0222

2

xEmdxd

Region II:

xExVdxd

m o

2

22

2

xEVmdxd

o

22

2 2

22 2

mE

2

2 2

EVml o

E < 0-Vo

The solutions are, in general:

•Region I:                                                

•Region II:                                                         

•Region III:                                              In these the A's, B's and C's are constants.

Potential is even function, therefore the solutions can be even or odd. For even solution:

Κ and l are functions of E, so is formula for allowed energies.

Bound States of the Square Well

22 2

mE

2o2 EVm2l

222 2

omV

22 zza o 1

2

2

z

zztan o

Wide, Deep WellIf z is very large,

Infinite square well energies for well width of 2a.

This is half the energy, the others come from the odd wave functions.

12

2

z

zztan o

Shallow Narrow Well

As zo decreases, fewer bound states, until finally, )for zo < the lowest odd state disappears) only one bound state remains.

There is one bound state no matter how weak the well becomes.

12

2

z

zztan o

Finite Well Energy Levels

The energy levels for an electron in a potential well of depth 64 eV and width 0.39 nm are shown in comparison with the energy levels of an infinite well of the same size.

Particle in Finite-Walled Box

For the finite potential well, the solution to the Schrodinger equation gives a wavefunction with an exponentially decaying penetration into the classicallly forbidden region. Confining a particle to a smaller space requires a larger confinement energy. Since the wavefunction penetration effectively "enlarges the box", the finite well energy levels are lower than those for the infinite well.

PROBLEM 1

PROBLEM 2