Hamiltonian Problem (Quantum Mechanics)

[Just_some_guy]

Hi everyoneI have been give a matrix operator and asked to find the eigen values, I have done so and then I was given a state |ψ> of some particle.

The part I'm struggling with is it then asks for <H>, the expectation value of the matrix operator. It's a 3x3 matrix also.

I've tried using the generalised formula for expectation value but I've only ever used it for simple probability densities

I've looked all over the internet to find an example even a little close and had no joy whatsoever :(

Any ideas or guidance would be much appreciated :)Regards

 

[ShayanJ]

Hi
You should give us enough information and also show some attempt.

 

[Just_some_guy]

Above is the only thing I'm unsure about! Does the hermitian conjugate of the include the constant or not? Other than that I think I've solved the problem?
Thanks

 

[ShayanJ]

Yeah, that's right!

 

[paranormal]

,
Hello,

The Hamiltonian problem in quantum mechanics involves finding the eigenvalues and eigenvectors of a given matrix operator, which represents the total energy of a quantum system. The expectation value of the Hamiltonian operator, denoted as <H>, is the average energy of the system when it is in the state |ψ>. To calculate this, you can use the general formula for expectation value, which involves taking the inner product of the state vector and the Hamiltonian operator.

In your case, as the Hamiltonian operator is a 3x3 matrix, you will need to perform matrix multiplication to find the expectation value. Make sure to use the conjugate transpose of the state vector, denoted as <ψ|, in the calculation. If you are still having trouble, you can consult your textbook or a reliable online resource for a step-by-step example.

I hope this helps. Good luck with your calculations!

Best,