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Quantum Engi Q and A_Part_13

# Quantum Engi Q and A_Part_13 - 3.6 THE HARMONIC OSCILLATOR...

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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 ψ 100 + ψ 010 2 = x/ℓ ( πℓ 2 ) 3 / 4 e x 2 y 2 + z 2 ) / 2 2 . which is exactly the same as ψ 100 above, except in terms of ¯ x and ¯ y . So it is ψ 100 in the rotated frame. The other combination goes the same way. 3.6.6 Non-eigenstates

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38 CHAPTER 3. BASIC IDEAS OF QUANTUM MECHANICS
Chapter 4 Single-Particle Systems
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Quantum Engi Q and A_Part_13 - 3.6 THE HARMONIC OSCILLATOR...

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