- #1

Woolyabyss

- 143

- 1

- Homework Statement
- Problem statement attached as image

- Relevant Equations
- Schrodinger equation

##U_1 \otimes U_2 = (1- i H_1 \ dt) \otimes (1- i H_2 \ dt)##

We can write ## | \phi_i(t) > \ = U_i(t) | \phi_i(0)>## where i can be 1 or 2 depending on the subsystem. The ## U ##'s are unitary time evolution operators.

Writing as tensor product we get

## |\phi_1 \phi_2> = (1- i H_1 \ dt) | \phi_1(0)> \otimes \ (1- i H_2 \ dt) | \phi_2(0)> ##

Since these states are normalised we may write

## 1 = < \phi_1 \phi_2 | \phi_1 \phi_2 > = < \phi_1(0)| 1 + H^2_1 dt^2 | \phi_1(0)> < \phi_2(0)| 1 + H^2_2 dt^2 | \phi_2(0)> ##

This is as far as I've gotten. Any help would be appreciated.

We can write ## | \phi_i(t) > \ = U_i(t) | \phi_i(0)>## where i can be 1 or 2 depending on the subsystem. The ## U ##'s are unitary time evolution operators.

Writing as tensor product we get

## |\phi_1 \phi_2> = (1- i H_1 \ dt) | \phi_1(0)> \otimes \ (1- i H_2 \ dt) | \phi_2(0)> ##

Since these states are normalised we may write

## 1 = < \phi_1 \phi_2 | \phi_1 \phi_2 > = < \phi_1(0)| 1 + H^2_1 dt^2 | \phi_1(0)> < \phi_2(0)| 1 + H^2_2 dt^2 | \phi_2(0)> ##

This is as far as I've gotten. Any help would be appreciated.