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9. REQUIRED Consider the one dimensional harmonic oscillator. (a) Prove that (b) Prove that (c) Prove that (d) Prove that for any eigenstate [n). H [a, a¹] = 1. = (a²a + 1/2) (n + 1 ² ) In). H|n) = ħw (n + (n|T|n) = (n|V|n) (8) (9) (10) (11)
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Transcribed Image Text:9. REQUIRED Consider the one dimensional harmonic oscillator. (a) Prove that (b) Prove that (c) Prove that (d) Prove that for any eigenstate [n). H [a, a¹] = 1. = (a²a + 1/2) (n + 1 ² ) In). H|n) = ħw (n + (n|T|n) = (n|V|n) (8) (9) (10) (11)
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Step 1: Determine the required quantities:
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Step 2: Determine the operators:
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Step 3: (a) Prove the first identity:
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Step 4: (b) Prove the Second identity:
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Step 5: (c) Prove the third identity:
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Step 6: (d) Prove the fourth identity:
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