PS5 - Physics 115A Problem Set 5 Due Tuesday October 27 5pm...

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Physics 115A, Problem Set 5 Due Tuesday, October 27 @ 5pm in the box outside Broida 1019 (PSR) Suggested reading: Griffiths 2.4. Study for the midterm! 1 The 3D Harmonic Oscillator Next quarter, when we encounter quantum systems in three spatial dimensions, we will often discover that we end up being able to write them as a one-dimensional problem! In fact, the stationary states of the 3D harmonic oscillator ( V = 1 2 2 r 2 for radial coordinate r ) are ψ ( r, θ, φ ) = u ( r ) r Y m ( θ, φ ) where u ( r ) solves an effective 1D Schr¨ odinger equation known as the Radial Equation : - ~ 2 2 m d 2 u dr 2 + V ( ) eff u = Eu with an effective 1D potential V ( ) eff = 1 2 2 r 2 + ~ 2 2 m ( + 1) r 2 where r 0 is the radial coordinate and is an arbitrary non-negative integer. We must find all u ( r ) that solve the radial equation for all possible non-negative integer values of . We could do this by solving the = 0 case, then the = 1 case, then the = 2 case, etc. But it would be much easier if we leave as some arbitrary non-negative integer and try to find the general solution. 1. Re-write the radial equation in terms of a useful dimensionless bookkeeping variable γ r (you get to pick the variable, but it should be dimensionless).
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  • Winter '03
  • Nelson
  • Physics, mechanics, Uncertainty Principle, non-negative integer, probability density, radial equation

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