00:01
In this question we have been given a harmonic oscillator is in the initial state which is given by si of x comma 0 is a phi not of x minus 3i a 5 1 of x where phi n of x are the energy eigen function of the oscillator in the first part we need to calculate the matrices xxxxxx cap and demetrizes p cap which represent the position and the momentum of the oscillator in this initial state correct so this is what we have been given this i will rewrite as a times of phi not of x minus three i times five one of x okay so just i have taken a common now what i will do my x cap will look like 1 minus 3 i 0 and 1 okay whereas my vector p cap sorry the matrix p cap representing the momentum it will be 1 over 3 minus 3 over root of 3 times of i 0 and 1 over root of 3 this is what i will get since we know that p cap is nothing but p over under root of m omega and here omega is one m is given to be equal to three okay so this is what we have been given now we got these things as well correct now in the next part what we have to find calculate the commutator of two matrices x -cab and p -cap okay now we have these two matrices we have to find out the commutator for the position and the momentum matrices.
02:09
Okay, so commutator is given by x -cap, p -cap, x -cab, x -cab is equal to minus x -cab -x -cab, p -cab, p -cab, which is equal to minus i.
02:30
So hence, the commutator is given by x -coma -p -cab, is equal to x -cap p -cap minus p -cap x -cap, correct? so first of all what we will get, we will try to find what is this x -cap p -cap.
02:59
So this will be 1 minus 3i, 0 ,1, and then p -cap is 1 over root 3, minus 3 over root 3, i, and here it is 0, 1 over root of 3.
03:12
Solving this i'm getting 1 over root of 3 6 over root 3 times i 0 1 over root 3 this is what i will get and then i am going to get p -cap x -cap which is 1 over under root of 3 minus 3 over under root of 3 and here it is 0 1 over root of 3 and then 1 minus 3 i 0 and 1 minus 3 i 0 and 1...