Calculate Expectation Value of Hamiltonian using Dirac Notation?

In summary, the conversation discusses the calculation of the expectation value of the Hamiltonian for a harmonic oscillator, given a normalized state |\psi> and the raising and lowering operators \hat{a} and \hat{a}^\dagger. The conversation explores solving the problem using both wave function and Dirac notation, with the suggestion to use the fact that the states are labeled by the eigenvalue of the number operator \hat{N} to simplify the calculation.
  • #1
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Homework Statement


I have the state:
[itex]|\psi>=cos(\theta)|0>+sin(\theta)|1>[/itex]
where [itex]\theta[/itex] is an arbitrary real number and [itex]|\psi>[/itex] is normalized.
And [itex]|0> and |1> refer to the ground state and first excited state of the harmonic oscillator.

Calculate the expectation value of the Hamiltonian for the harmonic oscillator.



Homework Equations


[itex]\hat{H}=\hbar*\omega(\hat{N}+1/2)[/itex]
where
[itex]\hat{N}=\hat{adagger}*\hat{a}[/itex]
The product of the raising and lowering operators

I also know that
[a,adagger]=1

The Attempt at a Solution


So far I know that I can solve this by converting the two states, 0 and 1, to the wave functions and solving the integral.
But I am curious as to how I can solve this using Dirac notation
[itex]<\hat{H}>=<\psi|\hat{H}|\psi>[/itex]

specifically I cannot figure out how to apply the derivatives in the momentum operator from N using this notation.
 
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  • #2
You can no doubt find the relationship between [itex]\hat{a}, \hat{a}^\dagger[/itex] and [itex]\hat{x},\hat{p}[/itex] in your textbook. However, there is a more direct way to solve the problem when you note that the states [itex]|n\rangle[/itex] are actually labeled by the corresponding eigenvalue of the number operator [itex]\hat{N}[/itex]. Use this fact to compute the value of [itex]\langle n|\hat{H}|m\rangle[/itex] and use that to compute [itex]\langle \psi|\hat{H}|\psi\rangle[/itex].
 

What is Dirac notation?

Dirac notation is a mathematical notation used in quantum mechanics to represent states, vectors, and operators. It was developed by physicist Paul Dirac and is also known as bra-ket notation.

What is the Hamiltonian operator?

The Hamiltonian operator is an operator in quantum mechanics that represents the total energy of a system. It is used to calculate the time evolution of a quantum state and is a sum of the kinetic and potential energy operators.

What is the expectation value of a Hamiltonian?

The expectation value of a Hamiltonian is the average value of the total energy of a system, calculated using the Hamiltonian operator. It represents the most likely value of the energy that will be measured in an experiment.

How do you calculate the expectation value of a Hamiltonian using Dirac notation?

To calculate the expectation value of a Hamiltonian using Dirac notation, you first need to express the Hamiltonian operator in terms of Dirac's bra-ket notation. Then, you take the inner product of the state vector with the Hamiltonian operator applied to that same state vector. This gives you the expectation value of the Hamiltonian for that particular state.

Why is calculating the expectation value of a Hamiltonian important?

Calculating the expectation value of a Hamiltonian is important because it allows us to predict the most likely outcomes of quantum mechanical experiments and provides insights into the behavior of quantum systems. It also helps us understand the energy levels and dynamics of a system, which is crucial for many applications in science and technology.

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