Exponential of hermitian matrix

[ZCOR]

Homework Statement

Let A be a Hermitian matrix and consider the matrix U = exp[-iA] defined by thr Taylor expansion of the exponential.

a) Show that the eigenvectors of A are eigenvectors of U. If the eigenvalues of A are a subscript(i) for i=1,...N, show that the eigenvalues of U are exp[-i*a subscript(i)].

b) Show that U is unitary.

This example is important in quantum mechanics when A=Ht/h, where H is the Hamiltonian operator, t is time, and h is Plank's constant. Then U evolves the wavefunction over a time t.

Homework Equations

I have uploaded what I have come up with so far, but not sure where to go with it

The Attempt at a Solution

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See uploaded file

 

[blue_leaf77]

You have $A\nu_i = a_i \nu_i$ where $a_i$ is the eigenvalue corresponding to an eigenvector $\nu_i$. Have you tried inputting this eigenvalue equation into the eigenvalue equation for $U$ using its Taylor expansion definition?

Show that U is unitary.

Do you know the property of the determinant of a unitary matrix?

 

[ZCOR]

Ok, will see if I can make that work. Thanks!

 

[ZCOR]

I'm trying to figure this out, but not sure how to input that equation into the Taylor expansion

 

[blue_leaf77]

Try applying $U$ in its Taylor form to an eigenvector of $A$
$$
(I + iA - \frac{1}{2}A^2 - i\frac{1}{6} A^3 + \ldots) \nu_i .
$$