1. CHAPTER 3 PROBLEMS IN ONE DIMENSION Particle in one dimensional box
Step Potential
Particle energy E less than potential height
Particle energy E greater than potential height
Potential Barrier (Quantum Mechanical Tunneling)

2. Particle in a 1-Dimensional Box
Classical Physics: The particle can exist anywhere in the box and follow a path in accordance to Newton’s Laws.
Quantum Physics: The particle is expressed by a wave function and there are certain areas more likely to contain the particle within the box.
V(x)=0
V(x)=∞ L x
Region I
Region II
Region III
V(x)= for L>x>0
V(x)=∞ for x≥L, x≤0 :

3. Time -Independent Schrödinger Equation is
Applying boundary conditions:
Region I and III:
Region II

4. Finding the Wave Function
Our new wave function:
But what is ‘A’?
This is similar to the general differential equation:
Normalizing wave function:
So we can start applying boundary conditions:
x=0 ψ=0
x=L ψ=0
where n is any integer
Calculating Energy Levels:
Since n is any integer
Our normalized wave function is:

5. The step potential (energy less than step height)
( II )
(free particle) Running wave
Exponential decay
Boundary Condition- 1

6. The wavefunction is Consider continuity of Ψ(x) at x=0
Boundary Condition 2
Consider continuity of dΨ(x)/dx at x=0
Boundary Condition 3
The wavefunction is

7. Reflection coefficient
The combination of an incident and a reflected wave of equal intensities to form a standing wave.

8. Exponential decay
Forbidden region
Running wave

9. Penetration depth
Penetration depth
Form uncertainty relation

10. Example 6-1. Estimate the penetration distance Dx for a very small dust particle, of radius r=10-6m and density r=104kg/m3, moving at the very low velocity v=10-2m/sec, if the particle impinges on a potential of height equal to twice its kinetic energy in the region to the left of the step.
Vo-E = K