Chapter 12 (III)

# Chapter 12 (III) - 12.6 Energy Levels and Quantum States ˆ...

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Unformatted text preview: 12.6 Energy Levels and Quantum States ). , ( ) , ( ˆ t r t r H j r r Ψ = Ψ ε In quantum mechanics, the N-particle system contained in a finite volume may exist in any one of an enormous number of discrete states determined by the Schrödinger equation Each energy eigen value ε j corresponds to one or more quantum states described by the wave function , that is, there may be many quantum states having the same energy. These states are called to be degenerate. The number of quantum states at an energy level ε j is called the degeneracy g j . ) , ( t r r Ψ Consider a simplest case in quantum mechanics: a particle of mass m in an one-dimensional box with infinitely high walls. In this case, the wave function is: The probability of finding the particle at a position x in the box is determined by ψ (x) = Asinkx (0 ≤ x ≤ L), and ψ (0)= ψ (L)=0. With the wavenumber k by n=1, 2, 3, … . ) ( ) , ( / h r t i e x t r ε ψ − = Ψ x=0 x=L V= ∞ V= ∞ , L n k π = This is analogous to the vibrational wave of a string with both ends fixed: ψ (x) = Asin(2 π x/ λ ), where k= 2 π / λ ....
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Chapter 12 (III) - 12.6 Energy Levels and Quantum States ˆ...

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