Quantum Physics Comes of Age

1. Quantum Physics Comes of Age I incident II transmitted reflected

2. The Schrödinger Equation Heisenberg’s Matrix Mechanics 1924: de Broglie suggests particles are waves • Mid-1925: Werner Heisenberg introduces Matrix Mechanics • Semi-philosophical, it only considers observable quantities • It used matrices, which were not that familiar at the time • It refused to discuss what happens between measurements • In 1927 he derives uncertainty principles • Late 1925: Erwin Schrödinger proposes wave mechanics • Used waves, more familiar to scientists at the time • Initially, Heisenberg’s and Schrödinger’s formulations were competing • Eventually, Schrödinger showed they were equivalent; different descriptions which produced the same predictions Both formulations are used today, but Schrödinger is easier to understand

3. The Free Schrödinger Equation • 1925: Erwin Schrödinger proposes wave mechanics • Peter Debye suggested to him he needed to find a wave equation for quantum mechanics • He hit on the idea of using complex waves • The rest is history • Starting point: Energy/Momentum relationship • Multiply by the wave function on the right • Use de Broglie relations to rewrite • Use relationships for complex waves to rewrite with derivatives

4. Sample Problem with Free Schrödinger Show that the following expression satisfies the free Schrödinger equation, and find the constant A:

5. Sample Problem with Free Schrödinger (2) Show that the following expression satisfies the free Schrödinger equation, and find the constant A: Multiply by 2mt5/2/

6. The Schrödinger Equation • The General Prescription for Classical  Quantum: • Write a formula for the energy in terms of momentum and position • Transform Energy and momentum using the following prescription: • Rewrite it as a wave equation • What if we have forces? • Need to add potential energy V(x,t) on top of kinetic energy term

7. Comments on Schrödinger Equation • 1. This equation is inherentlycomplex • You MUST use complexwave functions • 2. This equation is first order in time • It has only first derivatives with respect to time • If you know the value at t = 0, you can work it out at subsequent times • Proved using Taylor expansion: Initial conditions: Classical physics x(t = 0) and v(t = 0) Initial conditions: Quantum physics (x,t = 0)

8. The Superposition Principle • 3. This equation is linear • The wave functionappears to the first power everywhere • You can take linear combinations of solutions: Let 1 and 2 be two solutions of Schrödinger. Then so is where c1 and c2 are arbitrary complex numbers Q.E.D

9. Time Independent problems • Often [usually] thepotential does notdepend on time: V = V(x). • To solve this equation, we try separation of variable: • Plug this guess in: • Divide by theoriginal wave function • Note that left side is independent of x, and right side is independent of t. • Both sides must be independent of both x and t • Both sides must be equal to a constant, called E (the energy)

10. Solving the time equation • We have turned one equation into two • But the two equations are now ordinary differential equations • Furthermore, the first equation is easy to solve: • These types of solutions are called stationary states • Why? Don’t they have time in them? • The probability density is independent of time

11. The Time Independent Schrödinger Eqn Multiply by (x) again • This equation is much easier to solve than theoriginal equation • ODE’s are easier than PDE’s • It can pretty easily be solved numerically, if necessary • Note that it is a real equation – you don’t need complex numbers • Imagine finding all possible solutions n(x) with energy En • Then we can find solutions of the original Schrödinger Equation • The most general solution is superposition of this solution