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Problem set

# Problem set - Problem set-1.1 1(a Work out the wave...

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1 Problem set-1.1 1. (a) Work out the wave functions of Stark states of hydrogen atom for n=4 and m=0. (b) Graph the density in (r, θ) plane or using the two parabolic coordinates. (c) Use first order perturbation theory to calculate the energy shift for each state in a static electric field of strength 10 4 V/cm. Express your results in eV. Explain if you are justified to use the perturbation theory. (hint: You can construct the Stark states using spherical coordinates or parabolic coordinates.) (a)For n=4 and m=0 we have four Stark states (using n 1 +n 2 +|m|+1=n): n 1 n 2 0 3 1 2 2 1 3 0 Ψ 1 = |4030> = a 0 |4s>+a 1 |4p>+a 2 |4d>+a 3 |4f> Ψ 2 = |4120> = b 0 |4s>+b 1 |4p>+b 2 |4d>+b 3 |4f> Ψ 3 = |4210> = c 0 |4s>+c 1 |4p>+c 2 |4d>+c 3 |4f> Ψ 4 = |4300> = d 0 |4s>+d 1 |4p>+d 2 |4d>+d 3 |4f> where a i , b i , c i , and d i are Wigner 3-j coefficients: a b c d 0 1/2 1/2 1/2 1/2 1 3 / (2√5) 1/ (2√5) - 1/ (2√5) - 3 / (2√5) 2 1/2 -1/2 -1/2 1/2 3 1/ (2√5) - 3 / (2√5) 3 / (2√5) - 1/ (2√5) And the Stark states can be rewritten as: Ψ 1 = |4030> = 0.5|4s> + 3 / (2√5) |4p> + 0.5|4d> + 1/ (2√5) |4f> Ψ 2 = |4120> = 0.5|4s> + 1/ (2√5) |4p> - 0.5|4d> - 3 / (2√5) |4f> Ψ 3 = |4210> = 0.5|4s> -

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