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humanist rho
humanist rho is offline
Sep18-11, 08:39 AM
P: 95
1. The problem statement, all variables and given/known data

A particle of mass m is placed in the ground state of a one-dimensional harmonic
oscillator potential of the form

V(x)=1/2 kx2

where the stiffness constant k can be varied externally. The ground state wavefunction
has the form ψ(x)[itex]\propto[/itex] exp(−ax2[itex]\sqrt{k}[/itex]) where a is a constant. If, suddenly, the parameter k is changed to 4k, the probability that the particle will remain in the ground state of the new potential is;

(a) 0.47 (b) 0.06 (c) 0.53 (d) 0.67 (e) 0.33 (f) 0.94

2. The attempt at a solution

The system is in the ground state before changing k

ie, [itex]\int[/itex][itex]\Psi[/itex]*[itex]\Psi[/itex]dx = ([itex]\pi/2a\sqrt{k}[/itex])1/2 =1
When the parameter is changed;let the wave function be [itex]\Psi'[/itex]
the probability to be in ground state is;

[itex]\int[/itex][itex]\Psi'*[/itex][itex]\Psi'[/itex]dx = ([itex]\pi/4a\sqrt{k}[/itex])1/2 = [itex]\frac{1}{\sqrt{2}}[/itex][itex]\times[/itex]([itex]\pi/2a\sqrt{k}[/itex])1/2 =[itex]\frac{1}{\sqrt{2}}[/itex][itex]\times[/itex]1=0.707

But this is not there in the option.
Could anybody pls check the steps and tell me where's the mistake or correct it?
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