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Fund Quantum Mechanics Lect &amp; HW Solutions 47

# Fund Quantum Mechanics Lect &amp; HW Solutions 47 - x 2...

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3.6. THE HARMONIC OSCILLATOR 29 3.6.2.1 Solution harmb-a Question: Write out the ground state energy. Answer: Taking the generic expression E n x n y n z = 2 n x + 2 n y + 2 n z + 3 2 ¯ and substituting the lowest possible value, 0, for each of n x , n y , and n z , you get the ground state energy E 000 = 3 2 ¯ 3.6.2.2 Solution harmb-b Question: Write out the ground state wave function fully. Answer: Taking the generic expression ψ n x n y n z = h n x ( x ) h n y ( y ) h n z ( z ) and substituting n x = n y = n z = 0, you get the ground state eigenfunction ψ 000 = h 0 ( x ) h 0 ( y ) h 0 ( z ) . Now substitute for h 0 from table 3.1: ψ 000 = 1 ( πℓ 2 ) 3 / 4 e
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Unformatted text preview: x 2 / 2 ℓ 2 e − y 2 / 2 ℓ 2 e − z 2 / 2 ℓ 2 where the constant ℓ is as given in table 3.1. You can multiply out the exponentials: ψ 000 = 1 ( πℓ 2 ) 3 / 4 e − ( x 2 + y 2 + z 2 ) / 2 ℓ 2 . 3.6.2.3 Solution harmb-c Question: Write out the energy E 100 . Answer: Taking the generic expression E n x n y n z = 2 n x + 2 n y + 2 n z + 3 2 ¯ hω and substituting n x = 1, n y = n z = 0, you get E 100 = 5 2 ¯ hω...
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