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Finite Square Well Potential. For V=finite “outside” the well. Solutions to S.E. inside the well the same. Have different outside. The boundary conditions (wavefunction and its derivative continuous) give quantization for E<V 0 longer wavelength, lower Energy. Finite number of energy levels

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finite square well potential
Finite Square Well Potential
  • For V=finite “outside” the well. Solutions to S.E. inside the well the same. Have different outside. The boundary conditions (wavefunction and its derivative continuous) give quantization for E<V 0
  • longer wavelength, lower Energy. Finite number of energy levels
  • Outside:

P460 - square well

boundary condition
Boundary Condition
  • Want wavefunction and its derivative to be continuous
  • often a symmetry such that solution at +a also gives one at -a
  • Often can do the ratio (see book) and that can simplify the algebra

P460 - square well

finite square well potential3
Finite Square Well Potential

Equate wave function at boundaries

And derivative

P460 - square well

finite square well potential4
Finite Square Well Potential

E+R does algebra. 2 classes. Solve numerically

k1 and k2 both depend on E. Quantization sets

allowed energy levels

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finite square well potential5
Finite Square Well Potential

Number of bound states is finite. Calculate

assuming “infinite” well energies. Get n. Add 1

Electron V=100 eV width=0.2 nm

Deuteron p-n bound state. Binding energy 2.2 MeV

radius = 2.1 F (really need 3D S.E………)

P460 - square well

finite square well potential6
Finite Square Well Potential

Can do an approximation by guessing at the

penetration distance into the “forbidden” region. Use to estimate wavelength

Electron V=100 eV width=0.2 nm

d

P460 - square well

delta function potential
Delta Function Potential
  • d-function can be used to describe potential.
  • Assume attractive potential V and E<V bound state. Potential has strength l/a. Rewrite Schro.Eq

0

V

P460 - square well

delta function potential ii
Delta Function Potential II
  • Except for x=0 have exponential solutions.
  • Continuity condition at x=0 is (in some sense) on the derivative. See by integrating S.E in small region about x=0

0

V

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1d barriers square
1D Barriers (Square)
  • Start with simplest potential (same as square well)

V

0

0

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1d barriers e v
1D Barriers E<V
  • Solve by having wave function and derivative continuous at x=0

solve for A,B. As |y|2 gives probability or intensity

|B|2=intensity of plane wave in -x direction

|A|2=intensity of plane wave in +x direction

P460 - square well

1d barriers e v11
1D Barriers E<V
  • While R=1 still have non-zero probability to be in region with E<V

Electron with barrier V-E= 4 eV. What is approximate

distance it “tunnels” into barrier?

V

A

D

B

0

0

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1d barriers e v12
1D Barriers E>V
  • X<0 region same. X>0 now “plane” wave

Will have reflection and transmission at x=0. But

“D” term unphysical and so D=0

V

A

D

B

C

0

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1d barriers e v13
1D Barriers E>V
  • Calculate Reflection and Transmission probabilities. Note flux is particle/second which is |y|2*velocity

“same” if E>V. different k

Note R+T=1

R

T

1

0

E/V 1

P460 - square well

1d barriers example
1D Barriers:Example
  • 5 MeV neutron strikes a heavy nucleus with V= -50 MeV. What fraction are reflected? Ignore 3D and use simplest step potential.

5

R

0

-50 MeV

P460 - square well

1d barriers step e v
1D Barriers Step E<V
  • Different if E>V or E<V. We’ll do E<V. again solve by continuity of wavefunction and derivative

No “left” travelling wave

Incoming falling transmitted

reflected

0 a

P460 - square well

1d barriers step e v16
1D Barriers Step E<V

X=a

  • X=0

4 equations A,B,C,F,G (which can be complex)

A related to incident flux and is arbitrary. Physics in:

Eliminate F,G,B

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example 1d barriers step
Example:1D Barriers Step
  • 2 eV electron incident on 4 eV barrier with thickness: .1 fm or 1 fm

With thicker barrier:

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example 1d barriers step18
Example:1D Barriers Step
  • Even simpler

For larger V-E

larger k

More rapid decrease in wave function through

classically forbidden region

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bound states and tunneling
Bound States and Tunneling
  • State A will be bound with infinite lifetime. State B is bound but can decay to B B’+X (unbound) with lifetime which depends on barrier height and thickness.
  • Also reaction B’+XBA can be analyzed using tunneling
  • tunneling is ~probability for wavefunction to be outside well

B’

V

B

A

E=0

0

0

P460 - square well

alpha decay
Alpha Decay
  • Example: Th90  Ra88 + alpha
  • Kinetic energy of the alpha = mass difference
  • have V(r) be Coulomb repulsion outside of nucleus. But attractive (strong) force inside the nucleus. Model alpha decay as alpha particle “trapped” in nucleus which then tunnels its way through the Coulomb barrier
  • super quick - assume square potential
  • more accurate - 1/r and integrate

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alpha decay21
Alpha Decay

A B C

  • Do better. Use tunneling probability for each dx from square well. Then integrate
  • as V(r) is known, integral can be calculated. See E+R and Griffiths 8.2

4pZ=large number

P460 - square well

alpha decay22
Alpha Decay
  • T is the transmission probability per “incident” alpha
  • f=no. of alphas “striking” the barrier (inside the nucleus) per second = v/2R, If v=0.1c f=1021 Hz

Depends strongly on alpha kinetic energy

P460 - square well