1. Schrödinger Representation – Schrödinger Equation
Time dependent Schrödinger Equation
Developed through analogy to Maxwell’s equations and knowledge of
the Bohr model of the H atom.
Hamiltonian kinetic potential energy energy
Sum of kinetic energy and potential energy.
The potential, V, makes one problem different form another H atom, harmonic oscillator.
Q.M.
one dimension
three dimensions
recall
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2. Getting the Time Independent Schrödinger Equation
wavefunction
If the energy is independent of time
Try solution
product of spatial function and time function
Then
independent of t
independent of x, y, z
divide through by
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3. Can only be true for any x, y, z, t if both sides equal a constant.
depends only on t
depends only
on x, y, z
Can only be true for any x, y, z, t if both sides equal a constant.
Changing t on the left doesn’t change the value on the right. Changing x, y, z on right doesn’t change value on left.
Equal constant
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4. Both sides equal a constant, E.
Energy eigenvalue problem – time independent Schrödinger Equation
H is energy operator. Operate on f get f back times a number. f’s are energy eigenkets; eigenfunctions; wavefunctions.
E Energy Eigenvalues Observable values of energy
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5. Time Dependent Equation (H time independent)
Integrate both sides
Take initial condition at t = 0, F = 1, then C = 0.
Time dependent part of wavefunction for time independent Hamiltonian. Time dependent phase factor used in wave packet problem.
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6. Total wavefunction for time independent Hamiltonian.
E – energy (observable) that labels state.
Normalization
Total wavefunction is normalized if time independent part is normalized.
Expectation value of time independent operator S.
S does not depend on t, can be brought to other side of S.
Expectation value is time independent and depends only on the time independent part of the wavefunction.
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7. Expectation value of operator representing observable for state
Equation of motion of the Expectation Value in the Schrödinger Representation
Expectation value of operator representing observable for state
example - momentum
In Schrödinger representation, operators don’t change in time. Time dependence contained in wavefunction.
Want Q.M. equivalent of time derivative of a classical dynamical variable.
P P
classical momentum goes over to momentum operator
want Q.M. operator equivalent of time derivative
Definition: The time derivative of the operator , i. e., is defined to mean an operator whose expectation in any state is the time derivative of the expectation of the operator
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8. time dependent Schrödinger equation.
Want to find
Use
time dependent Schrödinger equation.
product rule
time independent – derivative is zero
Use the complex conjugate of the Schrödinger equation
(operate to left)
Then
, and from the Schrödinger equation
From complex conjugate of Schrödinger eq.
From Schrödinger eq.
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9. This is the commutator of H A.
Therefore
This is the commutator of H A.
The operator representing the time derivative of an observable is
times the commutator of H with the observable.
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10. Solving the time independent Schrödinger equation
The free particle
momentum problem
Free particle Hamiltonian – no potential – V = 0.
Commutator of H with P
commute Simultaneous Eigenfunctions.
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11. Free particle energy eigenvalue problem
Use momentum eigenkets.
energy eigenvalues
Therefore,
Energy same as classical result.
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12. Particle in a One Dimensional Box
Infinitely high, thick,
impenetrable walls
Particle inside box. Can’t get out because of impenetrable walls. Classically E is continuous. E can be zero. One D racquet ball court. Q.M.
E can’t be zero.
Schrödinger Equation
Energy eigenvalue problem
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13. 1. The wave function must be finite everywhere.
For
Want to solve differential Equation, but solution must by physically acceptable.
Born Condition on Wavefunction to make physically meaningful.
1. The wave function must be finite everywhere.
2. The wave function must be single valued.
3. The wave function must be continuous.
4. First derivative of wave function must be continuous.
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14. Functions with this property sin and cos.
Second derivative of a function equals a negative constant times the same function.
Functions with this property sin and cos.
These are solutions provided
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15. Solutions with any value of a don’t obey Born conditions.j = x
Well is infinitely deep. Particle has zero probability of being found outside the box.
 = 0 for
Function as drawn discontinuous at
To be an acceptable wavefunction
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16. –b b n is an integer Integral number of half wavelengths
in box. Zero at walls. b –b
Have two conditions for a2.
Solve for E.
Energy eigenvalues – energy levels, not continuous.
L = 2b – length of box.
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17. Energy levels are quantized. Lowest energy not zero.
L = 2b – length of box.
wavefunctions including
normalization constants
First few wavefunctions.
Quantization forced by Born conditions (boundary conditions)
Fourth Born condition not met – first derivative not continuous. Physically unrealistic problem because the potential is discontinuous.
Like classical string – has “fundamental” and harmonics.
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18. Particle in a Box Simple model of molecular energy levels.
Anthracene L
p electrons – consider “free” in box of length L. Ignore all coulomb interactions. E2 E1 S0 S1 DE
Calculate wavelength of absorption of light.
Form particle in box energy level formula
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19. Big molecules absorb in red. Small molecules absorb in UV.
Anthracene particularly good agreement. Other molecules, naphthalene, benzene, agreement much worse.
Important point Confine a particle with “size” of electron to box size of a molecule Get energy level separation, light absorption, in visible and UV.
Molecular structure, realistic potential give accurate calculation, but it is the mass and size alone that set scale.
Big molecules absorb in red. Small molecules absorb in UV.
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20. Particle in a Finite Box – Tunneling and Ionization
Box with finite walls.
Time independent Schrödinger Eq.
Inside Box V = 0
Second derivative of function equals negative constant times same function. Solutions – sin and cos.
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21. Two cases: Bound states, E < V Unbound states, E > V
Solutions inside box or
Outside Box
Two cases: Bound states, E < V Unbound states, E > V
Bound States
Second derivative of function equals positive constant times same function.
Not oscillatory.
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22. Then, solutions outside the box
Try solutions
Second derivative of function equals positive constant times same function.
Then, solutions outside the box
Solutions must obey Born Conditions
can’t blow up as
Therefore,
Outside box exp. decays Inside box oscillatory
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23. b = x j Outside box exp. decays Inside box oscillatory
The wavefunction and its first derivative
continuous at walls – Born Conditions. x
probability 0
Expanded view
For a finite distance into the material finite probability of finding particle.
Classically forbidden region V > E. b x j =
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24. Tunneling - Qualitative Discussionj = X
Classically forbidden region.
Wavefunction not zero at far side of wall.
Probability of finding particle
finite outside box.
A particle placed inside of box with not enough energy to go over the wall can Tunnel Through the Wall.
Formula derived in book
mass = me E = 1000 cm-1 V = 2000 cm-1
Wall thickness (d) 1 Å Å Å probability ratio  10-17
Ratio probs - outside vs. inside edges of wall.
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25. e-l V Chemical Reaction not enough energy to go over barrier
Temperature dependence of some chemical reactions shows to much product at low T. V > kT . V
Decay of probability in classically forbidden region for parabolic potential.
e-l
tunneling distance mass barrier height parameter
light particles tunnel
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26. H C H H Methyl Rotation Methyl groups rotate even at very low T.
Radioactive Decay
repulsive Coulomb interaction
nuclear attraction strong interaction
some probability outside nucleus
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27. E large enough - ionization
Unbound States and Ionization -b b
bound states
V(x) = V
If E > V - unbound states
E large enough - ionization
E < V
Inside the box (between -b and b) V = 0
Solutions oscillatory
Outside the box (x > |b|) E > V
Solutions oscillatory
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28. Wavefunction has equal amplitude everywhere.
unbound state
In limit
Wavefunction has equal amplitude everywhere.
To solve (numerically) Wavefunction and first derivative equal at walls, for example at x = b
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29. Continuous range of energies - free particle
For E >> V
for all x
As if there is no wall.
Continuous range of energies - free particle
Particle has been ionized.
Free particle wavefunction
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30. Tunneling Ionization In real world potential barriers are finite. 30
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