Fund Quantum Mechanics Lect & HW Solutions 55

Fund Quantum Mechanics Lect & HW Solutions 55 -...

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3.6. THE HARMONIC OSCILLATOR 37 The same way, you get ψ 010 = 2 y/ℓ ( πℓ 2 ) 3 / 4 e ( x 2 + y 2 + z 2 ) / 2 2 . So, the combination ( ψ 100 + ψ 010 ) / 2 is ψ 100 + ψ 010 2 = ( x + y ) /ℓ ( πℓ 2 ) 3 / 4 e ( x 2 + y 2 + z 2 ) / 2 2 . Now x 2 + y 2 + z 2 is according to the Pythagorean theorem the square distance from the origin, which is the same as ¯ x 2 + ¯ y 2 + z 2 . And since ¯ x = ( x + y ) / 2, the sum x + y in the combination eigenfunction above is x . So the combination eigenfunction is
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