Finding the probability of energy measurements

In summary, the conversation discusses how to calculate the probability of measuring specific outcomes when performing an energy measurement on a given system. The solution involves using the normalized wavefunction and the eigenstates corresponding to specific energies. The conversation concludes with the understanding that the probability can be calculated using the bracket notation.
  • #1
Denver Dang
148
1

Homework Statement


Hi... My problem says:

In a given experiment the system, at the time [itex]t = 0[/itex], in the normalized state is given by:

[tex]\phi(t =0) = \frac{1}{\sqrt{5}}(i\psi_{210} + 2\psi_{211})[/tex]

What possible outcomes is possible if you do an energy measurement on the system in this state, and with what probability does this occur ?


Homework Equations


I know that the states [itex]\psi_{21-1}, \psi_{210}[/itex] and [itex]\psi_{211}[/itex] is given by the matrix [itex]H[/itex]:

[tex]\[ \left( \begin{array}{ccc}
E_{2} + 2\gamma\hbar^{2} & 0 & 0 \\
0 & E_{2} + 2\gamma\hbar^{2} & 0 \\
0 & 0 & E_{2}\end{array} \right)\][/tex]

Don't know if I need to be telling more ?

The Attempt at a Solution


Well, I know the outcome has to be the two last states (In the order I wrote my states), which means: [itex]\psi_{210}[/itex] and [itex]\psi_{211}[/itex].
This means that I have to calculate the probability for those two, which means: [itex]P(E_{2} + 2\gamma\hbar^{2})[/itex] and [itex]P(E_{2})[/itex].

My problem is, that I'm not quite sure how to calculate that :/

My book says:

[tex]P(j) = \frac{N(j)}{N},[/tex]
but I have no idea how to make use of that in this case.

I know the answer should be: [itex]P(E_{2} + 2\gamma\hbar^{2}) = \frac{1}{5}[/itex] and [itex]P(E_{2}) = \frac{4}{5},[/itex]
but again, not sure how to do it.

So I was hoping someone could give me some pointers towards this, probably, easy question :)


Regards.
 
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  • #2
I will use the bracket notation...

You are given a normalized wavefunction [itex]|\Psi\rangle[/itex]. So the probability of measuring a specific eigenstate [itex]|\psi_j\rangle[/itex] is just:

[tex]P(j) = \left|\langle\psi_j|\Psi\rangle\right|^2[/tex]

Each eigenstate corresponds to a specific energy. So you know the probability of measuring a specific energy.
 
  • #3
Ahhh, ofc...
I think I have it now :)

Thank you very much.
 

1. What is the purpose of finding the probability of energy measurements?

The purpose of finding the probability of energy measurements is to understand the likelihood of a particular energy value occurring in a system. This can help in predicting the behavior of the system and making informed decisions.

2. How is the probability of energy measurements calculated?

The probability of energy measurements is calculated by dividing the number of energy values that fall within a certain range by the total number of possible energy values in the system.

3. What factors influence the probability of energy measurements?

The probability of energy measurements can be influenced by various factors such as the number of energy states in the system, the temperature of the system, and the energy distribution of the particles in the system.

4. Can the probability of energy measurements be greater than 1?

No, the probability of energy measurements cannot be greater than 1. It represents the likelihood of a particular energy value occurring and therefore must be between 0 and 1.

5. How is the probability of energy measurements related to the uncertainty principle?

The probability of energy measurements is related to the uncertainty principle, which states that it is impossible to know the exact energy of a system at a given time. The probability of energy measurements provides a range of possible energy values and reflects this uncertainty.

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