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 Oct. 1st
Read Chapters 4 and 5 of Kane
Chapter 4: 2, 3, 5, 13, 20
Chapter 5: 3, 4, 5, 7, 8

2. 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 = ±∞.

3. 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!

4. 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.

5. 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:

6. Reflection and Transmission
The wave function will consist of an incident wave, a reflected wave, and a transmitted wave.
Sines and cosines in all three regions
Since the wave moves from left to right, we can identify the solutions:
Incident wave
Reflected wave
Transmitted wave

7. Probability of Reflection and Transmission
The probability of the particle being reflected R or transmitted T is:
Because the particle must be either reflected or transmitted:
R + T = 1
By applying the boundary conditions x → ± ∞, x = 0, and x = L, we arrive at the transmission probability:
Note that the transmission probability can be 1.

8. 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:

9. 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.

10. 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.

11. The Scanning-Tunneling Microscope
Electrons must tunnel through the vacuum (barrier) from the surface to the tip. The probability is exponentially related to the distance, hence the ultrahigh resolution.
Image from IBM.
Image of a molecule