Chapt41_VG

Chapt41_VG - Chapter 41 1D Wavefunctions Chapter 41...

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Chapter 41 1D Wavefunctions

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Topics: Schrödinger’s Equation: The Law of Psi Solving the Schrödinger Equation A Particle in a Rigid Box: Energies and Wave Functions A Particle in a Rigid Box: Interpreting the Solution The Correspondence Principle Finite Potential Wells Wave-Function Shapes The Quantum Harmonic Oscillator More Quantum Models Quantum-Mechanical Tunneling Chapter 41. One-Dimensional Quantum Mechanics
The wave function is complex. i ! ! ! t " ( x , t ) = # ! 2 2 m ! 2 ! x 2 ( x , t ) + U ( x ) ( x , t ) What is the PDF for finding a particle at x ? P ( x , t ) = ! ( x , t ) 2 Step 1: solve Schrodinger equation for wave function Step 2: probability of finding particle at x is P ( x , t ) = ( x , t ) 2

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Stationary States - Bohr Hypothesis i ! ! ! t " ( x , t ) = # ! 2 2 m ! 2 ! x 2 ( x , t ) + U ( x ) ( x , t ) ! ( x , t ) = ˆ ( x ) e " 1 Et / ! E ˆ ( x ) = " ! 2 2 m # 2 # x 2 ˆ ( x ) + U ( x ) ˆ ( x ) = E ! Stationary State satisfies
E ˆ ! ( x ) = " ! 2 2 m # 2 # x 2 ˆ ( x ) + U ( x ) ˆ ( x ) Stationary states ! 2 ! x 2 ˆ " ( x ) = #\$ 2 ( x ) ˆ ( x ) Rewriting: 2 ( x ) = 2 m ! 2 E " U ( x ) ( ) Dependence on x comes from dependence on potential

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! 2 ! x 2 ˆ " ( x ) = #\$ 2 ( x ) ˆ ( x ) ! 2 ( x ) = 2 m ! 2 E " U ( x ) ( ) Requirements on wave function 1. Wave function is continuous 2. Wave function is normalizable E ! U ( x ) ( ) = K Classically K= kinetic energy
2 K ( )

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This note was uploaded on 12/28/2011 for the course PHYSICS 270 taught by Professor Drake during the Fall '08 term at Maryland.

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Chapt41_VG - Chapter 41 1D Wavefunctions Chapter 41...

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