Quantum Engi Q and A_Part_9

# So the one dimensional ground state energy ex1 is

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Unformatted text preview: en Ey1 and Ez1 are each 100 times larger than Ex1 . So the one-dimensional ground state energy Ex1 is smaller than the true ground state energy E111 = Ex1 + Ey1 + Ez1 by a factor 201. Which means it is oﬀ by 20,000%. 3.5.8.2 Solution pipeg-b Question: At what ratio of ℓy /ℓx does the energy E121 become higher than the energy E311 ? Answer: Using the given expression for Enx ny nz , Enx ny nz = ¯ ¯ n2 h2 π 2 n2 h2 π 2 n2 h2 π 2 x¯ + y 2 + z 2, 2 2mℓx 2mℓy 2mℓz E121 = E311 when 4¯ 2 π 2 h h2 π 2 ¯ 9¯ 2 π 2 h h2 π 2 ¯ h2 π 2 ¯ h2 π 2 ¯ + + = + + 2 2 2 2 2 2mℓx 2mℓy 2mℓz 2mℓx 2mℓy 2mℓ2 z 3.5. A PARTICLE CONFINED INSIDE A PIPE 27 Canceling the terms that both sides have in common: 3¯ 2...
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