Lect20_SHO2 - Lecture 20 Quantum SHO Part 2 Phy851 Fall...

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Lecture 20: Quantum SHO: Part 2 Phy851 Fall 2009
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Recap Introduced dimensionless variables: Introduce ‘normal variables’: Energy eigenvalues: Raising and lowering operators: λ X X = P P h = ω h H H = m h = 2 2 2 1 2 1 X P H + = ( ) P i X A + = 2 1 ( ) P i X A = 2 1 A , A [ ] = 1 2 1 + = A A H 1 = n c n A n 1 + = n d n A n ( ) n n n H 2 / 1 + = K , 3 , 2 , 1 , 0 = n 1 = n n 0 = n n
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Coefficients c n and d n Using n instead of ε , we have 1 = n c n A n 1 + = n d n A n 2 / 1 + = n n H n 2 / 1 2 / 1 + = + n n A A n n n A A n = n n A A n = n n n c n = 1 1 2 n c n = n n AA n = 1 1 + = n n AA n 1 1 1 2 + = + + n n n d n 1 + = n d n 1 = n n n A 1 1 + + = n n n A Must memorize
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How to Find Wavefunctions? Let us define: Let’s start simple and try to find the ground state wavefunction: An equation involving only | 0 is: We can try to use this somehow: We can write A in terms of X and P : Which gives: n x x n = ) ( ψ 0 ) ( 0 x x = 0 0 = A 0 0 = A x + = P i X A h λ 2 1 0 0 2 1 = + P i X x h 0 ) ( ) ( 0 0 = + x dx d x x
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Ground State Wavefunction
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Lect20_SHO2 - Lecture 20 Quantum SHO Part 2 Phy851 Fall...

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