View Single Post
humanist rho
humanist rho is offline
#1
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?
Phys.Org News Partner Science news on Phys.org
Simplicity is key to co-operative robots
Chemical vapor deposition used to grow atomic layer materials on top of each other
Earliest ancestor of land herbivores discovered