PS4 - Physics 115A Problem Set 4 Due Tuesday October 20 5pm...

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Physics 115A, Problem Set 4 Due Tuesday, October 20 @ 5pm in the box outside Broida 1019 (PSR) Suggested reading: Griffiths 2.3 1 Constructing States Given the explicit solution for the ground state of the harmonic oscillator, ψ 0 ( x ) = π ~ 1 / 4 e - 2 ~ x 2 1. Construct ψ 2 ( x ) explicitly using the algebra formalism. (Hint: ψ 1 ( x ) is constructed this way in Griffiths Example 2.4) 2. Construct ψ 3 ( x ) explicitly using the algebra formalism. 3. Find the classical turning points for each of these states (i.e., the values of x where the kinetic energy goes to zero for a classical harmonic oscillator of known total energy). 4. Sketch ψ 0 , ψ 1 , ψ 2 , ψ 3 . For each state, mark the classical turning point. How does the classical turning point relate to the form of the wavefunction? 2 Clever Commutation Suppose I give you a quantum system that can be described with a raising operator a + and a lowering operator a - that satisfy the commutation relation [ a - , a + ] = 1 (it doesn’t have to be the quantum harmonic oscillator, specifically). You know that there is a properly- normalized state ψ that is annihilated by a - , i.e., you know that a - ψ = 0, but I don’t tell you how a + acts on ψ . 1. I prepare for you a state Ψ = c × ( a - a - a - a - a + a + a + a + ψ ) where c is a real constant. Is Ψ a normalizable state? If so, find c such that Ψ is properly normalized state. What is the relation between Ψ and ψ ? 1
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2. I prepare for you a different state Ψ = c × ( a - a - a - a - a + a + a + ψ ) where c is a real constant. Is this new Ψ a normalizable state? If so, find c such that Ψ is properly normalized state. 3 Uncertainty Principle for the ground state Consider again the ground state of the harmonic oscillator. By explicit integration of the wavefunction, 1. Find h x i and h p i for this state. 2. Find h x 2 i and h p 2 i for this state. 3. Find the expectation value of the kinetic energy, h T i , and the expectation value of the potential energy, h V i , for this state. 4. Compute σ x σ p for this state, and check that the uncertainty principle is satisfied. 4 Uncertainty Principle for the n th state Consider the n th stationary state of the harmonic oscillator.
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