1. CHAPTER 6 Quantum Mechanics II
6.1 The Schrödinger Wave Equation
6.2 Expectation Values
6.3 Infinite Square-Well Potential
6.4 Finite Square-Well Potential
6.5 Three-Dimensional Infinite- Potential Well
6.6 Simple Harmonic Oscillator
6.7 Barriers and Tunneling
Erwin Schrödinger ( )
A careful analysis of the process of observation in atomic physics has shown that the subatomic particles have no meaning as isolated entities, but can only be understood as interconnections between the preparation of an experiment and the subsequent measurement.
- Erwin Schrödinger
Prof. Rick Trebino, Georgia Tech,

2. A superb book on the historical debates, issues, and personalities of the scientists who invented modern physics

3. Opinions on quantum mechanics
I think it is safe to say that no one understands quantum mechanics. Do not keep saying to yourself, if you can possibly avoid it, “But how can it be like that?” because you will get “down the drain” into a blind alley from which nobody has yet escaped. Nobody knows how it can be like that.
- Richard Feynman
Those who are not shocked when they first come across quantum mechanics cannot possibly have understood it.
- Niels Bohr
Richard Feynman ( )

4. Properties of Valid Wave Functions
Conditions on the wave function:
1. In order to avoid infinite probabilities, the wave function must be finite everywhere.
2. The wave function must be single valued.
3. The wave function must be twice differentiable. This means that it and its derivative must be continuous. (An exception to this rule occurs when V is infinite.)
4. In order to normalize a wave function, it must approach zero as x approaches infinity.
Solutions that do not satisfy these properties do not generally correspond to physically realizable circumstances.

5. Normalization and Probability
The probability P(x) dx of a particle being between x and x + dx is given in the equation
The probability of the particle being between x1 and x2 is given by
The wave function must also be normalized so that the probability of the particle being somewhere on the x axis is 1.

6. 6.2: Expectation Values
In quantum mechanics, we’ll compute expectation values. The expectation value, , is the weighted average of a given quantity. In general, the expected value of x is:
If there are an infinite number of possibilities, and x is continuous:
Quantum-mechanically:
And the expectation of some function of x, g(x):

7. Bra-Ket Notation
This expression is so important that physicists have a special notation for it.
The entire expression is called a bracket.
And is called the bra with the ket.
The normalization condition is then:

8. 6.1: The Schrödinger Wave Equation
The Schrödinger Wave Equation for a particle in a potential V(x,t) in one dimension is:
where
The Schrodinger Equation is the fundamental equation of Quantum Mechanics.
Note that it’s very different from the classical wave equation.
But, except for its inherent complexity (the i), it will have similar solutions.

9. General Solution of the Schrödinger Wave Equation when V = 0
Try the usual solution:
This works as long as:
which says that the total energy is the kinetic energy.

10. General Solution of the Schrödinger Wave Equation when V = 0
In free space (with V = 0), the wave function is:
which is a sine wave moving in the x direction.
Notice that, unlike classical waves, we are not taking the real part of this function. Y is, in fact, complex.
In general, the wave function is complex.
But the physically measurable quantities must be real.
These include the probability, position, momentum, and energy.

11. Time-Independent Schrödinger Wave Equation
The potential in many cases will not depend explicitly on time: V = V(x).
The Schrödinger equation’s dependence on time and position can then be separated. Let:
which yields:
Now divide by y(x) f(t):
The left side depends only on t, and the right side depends only on x. So each side must be equal to a constant. The time-dependent side is:

12. Time-Independent Schrödinger Wave Equation
Multiply both sides by f /iħ:
which is an easy differential equation to solve:
But recall our solution for the free particle:
in which f(t) = exp(-iwt), so: w = B / ħ or B = ħw, which means that: B = E !
So multiplying by y(x), the spatial Schrödinger equation becomes:

13. Stationary States The wave function can now be written as:
The probability density becomes:
The probability distribution is constant in time.
This is a standing-wave phenomenon and is called a stationary state.
Most important quantum-mechanical problems will have stationary-state solutions. Always look for them first.

14. Time-Independent Schrödinger Wave Equation
This equation is known as the time-independent Schrödinger wave equation, and it is as fundamental an equation in quantum mechanics as the time-dependent Schrodinger equation.
So physicists often write simply:
where:
is an operator yielding the total energy (kinetic plus potential energies).

15. Operators Operators are important in quantum mechanics.
All observables (e.g., energy, momentum, etc.) have corresponding operators.
The kinetic energy operator is:
Other operators are simpler, and some just involve multiplication.
The potential energy operator is just multiplication by V(x).

16. Momentum Operator
To find the operator for p, consider the derivative of the wave function of a free particle with respect to x:
With k = p / ħ we have:
This yields:
This suggests we define the momentum operator as:
The expectation value of the momentum is:

17. Position and Energy Operators
The position x is its own operator. Done.
Energy operator: Note that the time derivative of the free-particle wave function is:
Substituting w = E / ħ yields:
This suggests defining the energy operator as:
The expectation value of the energy is:

18. Deriving the Schrödinger Equation using operators
The energy is:
Substituting operators:
E :
K+V :

19. Solving the Schrödinger Equation when V is constant.
Rearranging:
When V0 > E:
where:
Because the sign of the constant a2 is positive, the solution is:
Sometimes people use:
When E > V0:
where:
Because the sign of the constant -k2 is negative, the solution is:

20. 6.3: Infinite Square-Well Potential
Consider a particle trapped in a box with infinitely hard walls that the particle cannot penetrate. This potential is called an infinite square well and is given by: L x
Outside the box, where the potential is infinite, the wave function must be zero.
Inside the box, where the potential is zero, the energy is entirely kinetic, so E > 0 = V0:
So, inside the box, the solution is:
Taking A and B to be real.
where

21. Quantization and Normalization
Boundary conditions dictate that the wave function must be zero at x = 0 and x = L. This yields solutions for integer values of n such that kL = np.
The wave functions are: x L
The same functions as those for a vibrating string with fixed ends!
½ - ½ cos(2npx/L)
In QM, we must normalize the wave functions:
The normalized wave functions become:

22. Quantized Energy We say that k is quantized:
Solving for the energy yields:
The energy also depends on n. So the energy is also quantized.
The special case of n = 1 is called the ground state.

23. 6.4: Finite Square-Well Potential
Assume: E < V0
The finite square-well potential is:
The solution outside the finite well in regions I and III, where E < V0, is:
Realizing that the wave function must be zero at x = ±∞.

24. Finite Square-Well Solution (continued)
Inside the square well, where the potential V is zero, the solution is:
Now, the boundary conditions require that:
So the wave function is smooth where the regions meet.
Note that the wave function is nonzero outside of the box!

25. The particle penetrates the walls!
This violates classical physics!
The penetration depth is the distance outside the potential well where the probability decreases to about 1/e. It’s given by:
Note that the penetration distance is proportional to Planck’s constant.

26. Tunneling
Consider a particle of energy E approaching a potential barrier of height V0, and the potential everywhere else is zero. Also E < V0.
The solutions in Regions I and III are sinusoids.
The wave function in region II becomes:
Enforcing continuity at the boundaries:

27. Tunneling wave function
The particle can tunnel through the barrier—even though classically it doesn’t have enough energy to do so!
The transmission probability for tunneling is:
The uncertainty principle allows this violation of classical physics.
The particle can violate classical physics by DE for a short time, Dt ~ ħ / DE.

28. Analogy with Wave Optics
If light passing through a glass prism reflects from an internal surface with an angle greater than the critical angle, total internal reflection occurs.
However, the electromagnetic field isn’t exactly zero just outside the prism.
If we bring another prism very close to the first one, light passes into the second prism.
The is analogous to quantum- mechanical tunneling.

29. Alpha-Particle Decay
Tunneling explains alpha-particle decay of heavy, radioactive nuclei.
Outside the nucleus, the Coulomb force dominates.
Inside the nucleus, the strong, short-range attractive nuclear force dominates the repulsive Coulomb force. The potential is ~ a square well.
The potential barrier at the nuclear radius is several times greater than the energy of an alpha particle.
In quantum mechanics, however, the alpha particle can tunnel through the barrier.
This is radioactive decay!

30. 6.5: Three-Dimensional Infinite-Potential Welly
The wave function must be a function of all three spatial coordinates.
So consider momentum as an operator in three dimensions: x z
The three-dimensional Schrödinger wave equation is:
or:

31. The 3D infinite potential well
It’s easy to show that:
where:
and: x z y Lx Lz Ly
When the box is a cube:

32. Degeneracy
Try 10, 4, 3 and 8, 6, 5
Note that more than one wave function can have the same energy.
When more than one wave function has the same energy, those quantum states are said to be degenerate.
Degeneracy results from symmetries of the potential energy function that describes the system.
A perturbation of the potential energy can remove the degeneracy. Examples of perturbations include external electric or magnetic fields or various internal effects, like the magnetic fields due to the spins of the various particles.

33. 6.6: Simple Harmonic Oscillator
Simple harmonic oscillators describe many physical situations: springs, diatomic molecules and atomic lattices.
Consider the Taylor expansion of an arbitrary potential function:
Near a minimum, V1[x-x0] ≈ 0.

34. Simple Harmonic Oscillator
Consider the second-order term of the Taylor expansion of a potential function:
Letting x0 = 0.
Substituting this into Schrödinger’s equation:
We have:
Let and , which yields:

35. The Parabolic Potential Well
The wave function solutions are: where Hn(x) are Hermite polynomials of order n.

36. The Parabolic Potential Wellyn
|yn |2
The Parabolic Potential Well
Classically, the probability of finding the mass is greatest at the ends of motion (because its speed there is the slowest) and smallest at the center.
Contrary to the classical one, the largest probability for the lowest energy states is for the particle to be at (or near) the center.
Classical result

37. Correspondence Principle for the Parabolic Potential Well
As the quantum number (and the size scale of the motion) increase, however, the solution approaches the classical result. This confirms the Correspondence Principle for the quantum-mechanical simple harmonic oscillator.
Classical result

38. The Parabolic Potential Well
The energy levels are given by:
The zero point energy is called the Heisenberg limit: