00:01
The wave function as a function of x and at time t equal to 0 is given by a times sin 0x plus si 1 x where sin not x is the energy eigenstate with energy value e 0 and x x is the energy eigenstate with energy value e1.
00:20
Now since si not and si 1 are energy eigenstates, they must obey the orthormality conditions.
00:27
That means the norm of sin not and the norm of sine not and the norm of psi 1 equal to 1 and they are orthogonal to each other.
00:35
So if we integrate si not star x, x, x1x over x, we get 0.
00:40
Similarly, if we integrate si 1 star x, si not x over x, then that also gives us 0.
00:47
Now, in order to normalize the wave function si x 0, at tm equal to 0, we must have the integration of si star si over t x equal to 1.
01:01
That means if we apply the ortho -normality condition, we get mod of a squared, then si -0 star, si -0 plus si -1 star, si -1, integrated over dx equal to 1, because the cross terms of si -0 star, si -1, and si -1 star, psi -1, drops out by the orthonormality conditions.
01:35
This gives us mod of a square 1 plus 1 equal to 1 so mod of a is given by 1 by square root of 2 a can be any complex number so a must be equal to a can be any complex number whose modulus is given by 1 by square root of 2 so a is equal to 1 by square root of 2 e to the power i alpha where alpha is any real number next so we can just write this psi x not in the form e to the power i alpha square root 2 psi 0x plus si 1 x in part b, we have to find out the wave function x of t at a generic time t.
02:51
Now suppose si x x of x comma t is written as c0 t, si x0x plus c1 t, plus c1 t, but c0 t and c1 t are time dependent coefficients.
03:05
Now if we apply the hamiltonian operator, which is given by ih bar del l t, on both sides of this equation, we can find out the time evolution of this of this coefficient c0t and c1t...