1. The problem statement, all variables and given/known data
A particle of mass m is placed in the ground state of a onedimensional harmonic
oscillator potential of the form
V(x)=1/2 kx^{2}
where the stiffness constant k can be varied externally. The ground state wavefunction
has the form ψ(x)[itex]\propto[/itex] exp(−ax^{2}[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?
