Quick Quantum Eigenstate Question

[matt223]

Homework Statement

An atomic system has 2 alternative 2-state bases. The angular momentum bases are $$\left | \mu_i \right \rangle$$ with L_0 = 0 and L_1 = 1. The energy eigenstates are $$\left | \phi_i \right \rangle$$ with E_0 and E_1.

All states are normalised and:

$$\left | \mu_0 \right \rangle = \frac{1}{2} \left | \phi_0 \right \rangle + \frac{\sqrt 3}{2} \left | \phi_1 \right \rangle$$

Write down an expression for $$\left | \phi_1 \right \rangle$$ in terms of $$\left | \mu_i \right \rangle$$.

Homework Equations

The Attempt at a Solution

This should be straight forward however I cannot see how this can be done without an equation for $$\left | \mu_1 \right \rangle$$. Thank you for any help!

 

[ldamewood]

You may want to consider some of the basic properties of these quantum states. For example, what must the value of $$<\mu_1|\hat L|\mu_1>$$ be? What about $$<\mu_0|\hat L|\mu_1>$$? Is it possible to calculate the energy of the angular momentum states: $$<\mu_1|\hat H|\mu_1>$$?