Time evolution of quantum state with time ind Hamiltonian

[ianmgull]

Homework Statement

Part e)

Homework Equations

I know that the time evolution of a system is governed by a complex exponential of the hamiltonian:

|psi(t)> = Exp(-iHt) |psi(0)>

I know that |psi(0)> = (0, -2/Δ)

The Attempt at a Solution

I'm stuck on part e.

I was told by my professor that upon expanding the matrix exponential, I should get a familiar trig function. However I don't understand how this is possible.

Also, does this tell me that C+(t) will always be zero? Because the complex exponential multiplied by the first term in psi of zero is zero.

 

[TSny]

I know that |psi(0)> = (0, -2/Δ)

Would it be preferable to normalize this state vector?

I'm stuck on part e.

I was told by my professor that upon expanding the matrix exponential, I should get a familiar trig function. However I don't understand how this is possible.

Explicitly evaluate $H^2$, $H^3$, $H^4$, $H^5$,... Do enough of these to see the pattern.

Also, does this tell me that C+(t) will always be zero? Because the complex exponential multiplied by the first term in psi of zero is zero.

$e^{-iHt}$ is a matrix. This matrix operating on $\left( 0, \, 1 \right)^t$ will not necessarily produce a zero in the first entry of the output.

 

[ianmgull]

Thanks for the reply.

Your last point makes sense. I'm still having trouble wrapping my mind around matrix exponentials and forgot that the exponential would actually be a matrix.

I calculated the various powers of H like you mentioned. I definitely see a pattern: For Hⁿ, the individual elements are raised to the power n, and also divided by 2ⁿ. However I don't understand how to make sense of this pattern in a matrix exponential.

 

[TSny]

I definitely see a pattern: For Hⁿ, the individual elements are raised to the power n, and also divided by 2ⁿ. However I don't understand how to make sense of this pattern in a matrix exponential.

Can you describe what you got for $H^2$?
Note $H^2 = H H$, where $H H$ is matrix multiplication of $H$ times $H$.