HW9-sol - PHYSICS 4455 QUANTUM MECHANICS Problem Set 9 due...

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PHYSICS 4455 — QUANTUM MECHANICS Problem Set 9 — due 11 / 17 / 2005, in class. A few more items about harmonic oscillators. 1. Many of you wondered in class how the raising and lowering operator formalism connects back to the explicit wave functions in coordinate space. Here, we will see how this connection is established. An important step in the operator algebra involved the (well-founded) assumption that there had to be a ground state, | 0 , with ground state energy o = 1 / 2 in dimensionless form. When the lowering operator â acts on this state, the result is 0: Start from the equation â | 0 ⟩ = 0 (1) (i) Express â in terms of the operators x and p and write Eq. (1) in the position representation, as a differential equation for the ground state wave function ϕ o ( x ) . Solve this differential equation and normalize its solution properly. With the definitions â = m ω 2 x + i 1 2 m ω p â + = m ω 2 x i 1 2 m ω p and the identification p = − i d / dx , the differential equation becomes 0 = m ω 2 x + i 1 2 m ω i d dx ϕ o ( x ) d dx ϕ o ( x ) = − m ω x ϕ o ( x ) This can be solved by writing d ϕ o ϕ o = − m ω xdx ln ϕ o ( x ) = − m ω 2 x 2 + const which results in ϕ o ( x ) = C exp m ω 2 x 2 where C is a normalization constant. We can find it with the help of the usual Gaussian integral, −∞ +∞ dxe −α x 2 = π/α 1 = −∞ +∞ dx ϕ o ( x o ( x ) = C 2 −∞ +∞ dx exp m ω x 2 = C 2 π m ω C = m ω π 1 / 4 (ii) Next, we generate the first excited state, ϕ 1 ( x ) , from the ground state by using the raising operator property | 1 ⟩ = â + | 0 (2) Write â + in terms of the operators x and p and write Eq. (2) in the position representation.
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