At time t = 0, a particle of mass m in a 2-dimensional harmonic oscillator potential...

 

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At time t = 0, a particle of mass m in a 2-dimensional harmonic oscillator potential V(x, y) (x² + y²), = mw² 2 where w is a constant, is described by the wave function 2 p(x, y) = $20 (x, y) — ¹02(x, y), √3 where the eigenfunctions Pn,n,(x, y) have energy Enny = hw (1 + nx + ny). Write down the full time-dependent wave function (x, y, t). Is the asso- ciated probability distribution independent of time? Would this change if a small perturbing potential was applied that was i) a function of radius only or ii) a function of y only? [4 marks] At time t = 0, a particle of mass m in a 2-dimensional harmonic oscillator potential V(x, y) (x² + y²), = mw² 2 where w is a constant, is described by the wave function 2 p(x, y) = $20 (x, y) — ¹02(x, y), √3 where the eigenfunctions Pn,n,(x, y) have energy Enny = hw (1 + nx + ny). Write down the full time-dependent wave function (x, y, t). Is the asso- ciated probability distribution independent of time? Would this change if a small perturbing potential was applied that was i) a function of radius only or ii) a function of y only? [4 marks]