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lec29 - MIT OpenCourseWare http/ocw.mit.edu 8.044...

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MIT OpenCourseWare http://ocw.mit.edu 8.044 Statistical Physics I Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .
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Wavefunctions, One Particle s are the variables. r and r, ˆ H ( ˆ s ) Hamiltonian ˆ p, ˆ n is a state index and could Wavefunction ψ n ( s ) r, have several parts. For an e in hydrogen ψ = ψ n,l,m l ,m s ( s ) r, ˆ p, ˆ r, r, r, ˆ H ( ˆ s ) ψ n ( s ) = E n ψ n ( s ) 8.044 L29B1
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ψ n ( s ) often factors into space and spin parts. r, s ) = ψ space r ) ψ spin r, n ( n ( ψ n ( s ) ψ space ( x ) e αx 2 / 2 H n ( α x ) H.O. in 1 dimension n r ) e k · ψ space ( i r free particle in 3 dimensions n 8.044 L29B2
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ψ spin s n ( Spin is an angular momentum so for a given value of the magnitude S there are 2 S +1 values of m S For the case of S = 1 / 2 the eigenfunctions of the z s are φ 1 / 2 ( s ) component of s ) and φ 1 / 2 ( ¯ h S z φ 1 / 2 ( s ) ˆ s ) = φ 1 / 2 ( 2 ¯ h S z φ 1 / 2 ( s ) ˆ s ) = φ 1 / 2 ( 2 8.044 L29B3
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ψ spin s ) is not necessarily an eigenfunction of ˆ S z . For n (
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