Calculating Probability of Energy Measurement in Quantum Systems

[Axiom17]

Homework Statement

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

Homework Equations

[tex]|\psi\rangle=|1\rangle+i|2\rangle[/tex]

[itex]|1\rangle[/itex] is normalised stationary state with energy [itex]E_{1}[/itex]. Similarly with 2.

The Attempt at a Solution

I have the time-dep Schrodinger equation for [itex]\psi[/itex] as:

[tex]i \hbar \frac{\partial}{\partial t}|\psi \rangle = \hat{H}|\psi\rangle=E_{\psi}|\psi\rangle[/tex]

.. 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 [tex]|\psi>[/tex]?

 

[Axiom17]

For normalisation of [itex]|\psi\rangle[/itex] I calculated:

[tex]| |\psi\rangle |^{2}=\left(|1\rangle+i|2\rangle\right)\left(|1\rangle-i|2\rangle\right)=|1\rangle^{2}+|2\rangle^{2}[/tex]

 

[arkajad]

But what will be the normalized psi?

 

[Axiom17]

[tex]|\psi\rangle=E_{1}^{2}+E_{2}^{2}[/tex] ?

 

[Axiom17]

I've still not understood this

 

[arkajad]

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

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

What can you do with 1) knowing 2)?

 

[Axiom17]

that..

[tex]| |\psi\rangle |^{2}=E_{1}^{2}+E_{2}^{2}[/tex]

?

 

[Axiom17]

.. so

[tex]P(E_{1})=\frac{E_{1}^{2}}{E_{1}^{2}+E_{2}^{2}}[/tex]

? 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?