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

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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 ( θ, φ ) = r 3 8 π sin θe ± (1.1) Y 0 1 ( θ, φ ) = r 3 4 π cos θ (1.2) which allows us to write the Cartesian coordinates x, y, z as x = r sin θ cos φ = - r 2 π 3 r ( Y +1 1 ( θ, φ ) - Y - 1 1 ( θ, φ ) ) (1.3) y = r sin θ sin φ = i r 2 π 3 r ( Y +1 1 ( θ, φ ) + Y - 1 1 ( θ, φ ) ) (1.4) z = r cos θ = r 4 π 3 r Y 0 1 ( θ, φ ) . (1.5) Thus, we can rewrite the wavefunction ψ ( r, θ, φ ) = ( x + y + 3 z ) f ( r ) = ( i - 1) Y 1 1 ( θ, φ ) + ( i + 1) Y - 1 1 ( θ, φ ) + 3 2 Y 0 1 ( θ, φ ) ˜ f ( r )(1.6) where ˜ f ( r ) = p 2 π/ 3 rf ( r ). In Dirac notation this becomes | ψ i = h 3 2 | 1 , 0 i + ( i - 1) | 1 , 1 i + ( i + 1) | 1 , - 1 i i | ˜ f i . (1.7) Written in this form, it’s clear that | ψ 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

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2 Addition of 3 spin-1/2 States We’ll 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 we’re essentially done, since we know that the states must form complete repre- sentations of SU (2). The only possibility is a spin-3/2 state (with s z ∈ { 3 / 2 , 1 / 2 , - 1 / 2 , - 3 / 2 } ),
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342-PS1sol - Solutions to Problem Set 1 Physics 342 by...

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