Calculating Probability of Energy Measurement in Quantum Systems

[Axiom17]

Homework Statement

Quantum system in state $|\psi\rangle$. Energy of state measured at time $t$: Calculate probability that measurement will be $E_{1}$.

Homework Equations

$$|\psi\rangle=|1\rangle+i|2\rangle$$

$|1\rangle$ is normalised stationary state with energy $E_{1}$. Similarly with 2.

The Attempt at a Solution

I have the time-dep Schrödinger equation for $\psi$ as:

$$i \hbar \frac{\partial}{\partial t}|\psi \rangle = \hat{H}|\psi\rangle=E_{\psi}|\psi\rangle$$

.. but that's it.

I really don't know where to start with this

 

[arkajad]

First of all: To calculate probabilities you always need to normalize the states. While |1> and |2> are normalized by assumptions, what about $$|\psi>$$?

 

[Axiom17]

For normalisation of $|\psi\rangle$ I calculated:

$$| |\psi\rangle |^{2}=\left(|1\rangle+i|2\rangle\right)\left(|1\rangle-i|2\rangle\right)=|1\rangle^{2}+|2\rangle^{2}$$

 

[arkajad]

But what will be the normalized psi?

 

[Axiom17]

$$|\psi\rangle=E_{1}^{2}+E_{2}^{2}$$ ?

 

[Axiom17]

I've still not understood this

 

[arkajad]

You wrote:
1)
$$| |\psi\rangle |^{2}=\left(|1\rangle+i|2\rangle\right)\left(|1\rangle-i|2\rangle\right)=|1\rangle^{2}+|2\rangle^{2}$$But before that:

2) |1> is normalised stationary state with ... Similarly with 2.

What can you do with 1) knowing 2)?

 

[Axiom17]

that..

$$| |\psi\rangle |^{2}=E_{1}^{2}+E_{2}^{2}$$

?

 

[Axiom17]

.. so

$$P(E_{1})=\frac{E_{1}^{2}}{E_{1}^{2}+E_{2}^{2}}$$

? or something like that

 

[arkajad]

So you did not read about normalization of states. And you should! That's bad!

Which is your basic textbook?