Manfra17-2011

Manfra17-2011 - 6.1 The Schrödinger Wave Equation 6.2...

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Unformatted text preview: 6.1 The Schrödinger Wave Equation 6.2 Expectation Values 6.3 Infinite Square-Well Potential 6.4 Finite Square-Well Potential 6.5 Three-Dimensional Infinite- Potential Well 6.6 Simple Harmonic Oscillator 6.7 Barriers and Tunneling CHAPTER 6 Quantum Mechanics II Erwin Schrödinger (1887-1961) Note: 2nd exam is now scheduled for Wednesday March 23rd . Problem set due this coming Monday. The potential in many cases will not depend explicitly on time. The dependence on time and position can then be separated in the Schrödinger wave equation. Let: which yields: Now divide by the wave function ψ ( x ) f ( t ) : Time-Independent Schrödinger Wave Equation The left side depends only on t , and the right side depends only on x . So each side must be equal to a constant. The time dependent side is: Multiply both sides by f ( t )/ iħ : which is an easy differential equation to solve. Time-Independent Schrödinger Wave Equation But recall our solution for the free particle: in which f ( t ) = exp(-i ϖ t ) , so: ϖ = B / ħ or B = ħ ϖ , which means that: B = E ! So multiplying by ψ ( x ) , the spatial Schrödinger equation becomes: / ( ) e iEt f t- = h / f B f i t ∂ = ∂ h / / ( ) e e Bt i iBt f t- = = h h ( 29 ( , ) e i kx t x t ϖ- Ψ = ignoring the proportionality constant, which will come from the normalization condition Time-Independent Schrödinger Wave Equation This equation is known as the time-independent Schrödinger wave equation , and it is as fundamental an equation in quantum mechanics as...
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This note was uploaded on 04/23/2011 for the course PHYS 342 taught by Professor Staff during the Spring '08 term at Purdue.

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Manfra17-2011 - 6.1 The Schrödinger Wave Equation 6.2...

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