1. CHAPTER 5 The Schrodinger Eqn.
5.1 The Schrödinger Wave Equation
5.2 Expectation Values
5.3 Infinite Square-Well Potential
5.4 Finite Square-Well Potential
5.5 Three-Dimensional Infinite- Potential Well
5.6 Simple Harmonic Oscillator
5.7 Barriers and Tunneling
Erwin Schrödinger ( )
Homework due next Wednesday Sept. 30th
Read Chapters 4 and 5 of Kane
Chapter 4: 2, 3, 5, 13, 20 problems
Chapter 5: 3, 4, 5, 7, problems

2. 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.

3. Operators
The time-independent Schrödinger wave equation is as fundamental an equation in quantum mechanics as the time-dependent Schrödinger equation.
So physicists often write simply:
where:
is an operator yielding the total energy (kinetic plus potential energies).

4. 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).

5. 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:

6. 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:

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

8. Operators and Measured Values
In any measurement of the observable associated with an operator A, the only values that can ever be observed are the eigenvalues. Eigenvalues are the possible values of a in the Eigenvalue Equation: ˆ
where a is a constant and the value that is measured.
For operators that involve only multiplication, like position and potential energy, all values are possible.
But for others, like energy and momentum, which involve operators like differentiation, only certain values can be the results of measurements. In this case, the function Y is often a sum of the various wave function solutions of Schrödinger’s Equation, which is in fact the eigenvalue equation for the energy operator.

9. 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:

10. 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,
E>V0
So, inside the box, the solution is:
Taking A and B to be real.
where

11. 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:

12. 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.

13. 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 = ±∞.

14. 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!

15. 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.

16. Barriers and Tunneling
Consider a particle of energy E approaching a potential barrier of height V0, and the potential everywhere else is zero and E > V0.
In all regions, the solutions are sine waves. In regions I and III, the values of k are:
In the barrier region: