A question about energy in a harmonic potential

[Physicist]

Hi all,

If there is a particle in a harmonic potentail how can we find the probability that a single measurment of the system would yield to a certain energy?

How can we know which is the most probable enegy?

Thanks

 

[somy]

I think you can find a neat answer in all elementary quantum physisics books.
See for example:
Griffiths

 

[quantumdude]

You need to know the state of the particle as a linear combination of the eigenstates φ_n of the SHO Hamiltonian:

ψ=Σa_nφ_n

The probability of finding the partcle with energy E_n is the square of the modulus of the corresponding a_n, and the most probable energy is the eigenvalue for which this probability is largest.

 

[Physicist]

and the most probable energy is the eigenvalue for which this probability is largest.

Should I calculate the probability for all possible energies to know which one has the largest probability??

Thanks

 

[quantumdude]

Should I calculate the probability for all possible energies to know which one has the largest probability??

If you can't tell by inspection, then you'll have to.

 

[Physicist]

Can I say that the expectation value of the energy always has the largest probability?

&

In which cases can I tell by inspection? Can you give an example please?

Thank you very much..

 

[jujio77]

If you want the most probably Energy or whatever just think about it this way. The probability density for some operator X, is just

psi^* X psi.

Find the maximum for this distribution.