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incident transmitted reflected III

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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

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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

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

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

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The Schrödinger Equation What if we have forces? Need to add potential energy V(x,t) on top of kinetic energy term The General Prescription for Classical Quantum: 1.Write a formula for the energy in terms of momentum and position 2.Transform Energy and momentum using the following prescription: 3.Rewrite it as a wave equation

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Comments on Schrödinger Equation 1. This equation is inherently complex You MUST use complex wave 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)

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The Superposition Principle 3. This equation is linear The wave function appears 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 c 1 and c 2 are arbitrary complex numbers Q.E.D

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Time Independent problems Often [usually] the potential does not depend on time: V = V(x). To solve this equation, we try separation of variable: Plug this guess in: Divide by the original 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)

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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

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The Time Independent Schrödinger Eqn Multiply by (x) again This equation is much easier to solve than the original 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 E n Then we can find solutions of the original Schrödinger Equation The most general solution is superposition of this solution

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