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- Homework Statement
- Given a Hamiltonian ##H = \hbar \omega \sigma_1##, you are supposed to find the associated time-evolution unitary operator ##U(t)##.

- Relevant Equations
- Time-independent Schrodinger's equation

$$-i \hbar \frac{\partial | \psi (t) \rangle}{\partial t} = \hat{H} | \psi (t) \rangle$$

The associated unitary operator is

$$U(t) = \exp (\frac{-i \hat{H} t}{\hbar})$$

Now from the relevant equations,

$$U(t) = \exp(-i \omega \sigma_1 t)$$

which is easy to compute provided the Hamiltonian is diagonalized. Writing ##\sigma_1## in its eigenbasis, we get

$$\sigma_1 =

\begin{pmatrix}

1 & 0\\

0 & -1\\

\end{pmatrix}

$$

and hence the unitary ##U(t)## becomes

$$U(t) =

\begin{pmatrix}

e^{-i \omega t} & 0\\

0 & e^{i \omega t}\\

\end{pmatrix}

$$

Mind you that the above representation of $U(t)$ is in the basis ##\{ |+\rangle, |-\rangle\}## where

$$ |+\rangle =

\frac{1}{\sqrt{2}}

\begin{pmatrix}

1\\

1\\

\end{pmatrix}

$$

$$ |-\rangle =

\frac{1}{\sqrt{2}}

\begin{pmatrix}

1\\

-1\\

\end{pmatrix}

$$

Now, I need to write ##U(t)## back in the original basis ##\{|0\rangle, |1\rangle\}## (which is where I'm facing an issue). Finding the components of the above expression for ##U(t)## in the original basis,

$$\langle 0 | U(t) | 0 \rangle = e^{-i \omega t} \qquad \langle 1 | U(t) | 1 \rangle = e^{i \omega t}$$

with ##\langle 0 | U(t) | 1 \rangle = 0## and ##\langle 1 | U(t) | 0 \rangle = 0##. This gives me the exact same matrix representation in the original basis. Obviously this is not true and I'm doing something wrong.

$$U(t) = \exp(-i \omega \sigma_1 t)$$

which is easy to compute provided the Hamiltonian is diagonalized. Writing ##\sigma_1## in its eigenbasis, we get

$$\sigma_1 =

\begin{pmatrix}

1 & 0\\

0 & -1\\

\end{pmatrix}

$$

and hence the unitary ##U(t)## becomes

$$U(t) =

\begin{pmatrix}

e^{-i \omega t} & 0\\

0 & e^{i \omega t}\\

\end{pmatrix}

$$

Mind you that the above representation of $U(t)$ is in the basis ##\{ |+\rangle, |-\rangle\}## where

$$ |+\rangle =

\frac{1}{\sqrt{2}}

\begin{pmatrix}

1\\

1\\

\end{pmatrix}

$$

$$ |-\rangle =

\frac{1}{\sqrt{2}}

\begin{pmatrix}

1\\

-1\\

\end{pmatrix}

$$

Now, I need to write ##U(t)## back in the original basis ##\{|0\rangle, |1\rangle\}## (which is where I'm facing an issue). Finding the components of the above expression for ##U(t)## in the original basis,

$$\langle 0 | U(t) | 0 \rangle = e^{-i \omega t} \qquad \langle 1 | U(t) | 1 \rangle = e^{i \omega t}$$

with ##\langle 0 | U(t) | 1 \rangle = 0## and ##\langle 1 | U(t) | 0 \rangle = 0##. This gives me the exact same matrix representation in the original basis. Obviously this is not true and I'm doing something wrong.

Last edited: