342-PS1sol - Solutions to Problem Set 1 Physics 342 by:...

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Unformatted text preview: Solutions to Problem Set 1 Physics 342 by: Callum Quigley 1 Wavefunctions and Spherical Harmonics Recall the definitions of the = 1 spherical harmonic functions: Y 1 1 ( , ) = r 3 8 sin e i (1.1) Y 1 ( , ) = r 3 4 cos (1.2) which allows us to write the Cartesian coordinates x,y,z as x = r sin cos =- r 2 3 r ( Y +1 1 ( , )- Y- 1 1 ( , ) ) (1.3) y = r sin sin = i r 2 3 r ( Y +1 1 ( , ) + Y- 1 1 ( , ) ) (1.4) z = r cos = r 4 3 r Y 1 ( , ) . (1.5) Thus, we can rewrite the wavefunction ( r,, ) = ( x + y + 3 z ) f ( r ) = ( i- 1) Y 1 1 ( , ) + ( i + 1) Y- 1 1 ( , ) + 3 2 Y 1 ( , ) f ( r )(1.6) where f ( r ) = p 2 / 3 rf ( r ). In Dirac notation this becomes | i = h 3 2 | 1 , i + ( i- 1) | 1 , 1 i + ( i + 1) | 1 ,- 1 i i | f i . (1.7) Written in this form, its clear that | i is a sum of = 1 states, with m = 0 , 1 components. Then we have the following Measurement Probability L 2 = 2 ~ 2 1 L z = ~ 1 / 11 L z = 0 9 / 11 L z =- ~ 1 / 11 (1.8) 1 2 Addition of 3 spin-1/2 States Well use the notation | i to denote the states in the S 1 z S 2 z S 3 z basis. It is then trivial to compute the S z = i S iz eignevalues: State S z State S z | + + + i + 3 2 ~ |- - + i - 1 2 ~ | + +-i + 1 2 ~ |- +-i - 1 2 ~ | +- + i + 1 2 ~ | +- -i - 1 2 ~ |- + + i + 1 2 ~ |- - -i - 3 2 ~ (2.1) At this point were essentially done, since we know that the states must form complete repre-...
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342-PS1sol - Solutions to Problem Set 1 Physics 342 by:...

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