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)

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)=0 for L>x>0 V(x)=∞ for x≥L, x≤0 :

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

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:

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

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

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

Exponential decay Forbidden region Running wave

Penetration depth Penetration depth Form uncertainty relation

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