What is the Hamiltonian for a harmonic oscillator?

One of the most important problems in quantum mechanics is the simple harmonic oscillator, in part because its properties are directly applicable to field theory. , puts the Hamiltonian in the form H = p2 2m + mω2×2 2 resulting in the Hamiltonian operator, ˆH = ˆP2 2m + mω2 ˆX2 2 We make no choice of basis.

What does it mean to Diagonalize a Hamiltonian?

Exact diagonalization (ED) is a numerical technique used in physics to determine the eigenstates and energy eigenvalues of a quantum Hamiltonian. In this technique, a Hamiltonian for a discrete, finite system is expressed in matrix form and diagonalized using a computer.

How do you find the energy of a Hamiltonian matrix?

Given a Hamiltonian, the possible energy levels correspond to the eigenvalues of the Hamiltonian (no “plugging in” needed). More precisely, we have H|ψ⟩=E|ψ⟩ for every eigenvector. Given a normalized eigenvector, you can find the probability by ⟨ψ|H|ψ⟩, otherwise you have to normalize (divide by ⟨ψ|ψ⟩).

How do you solve Hamiltonian equations?

We can solve Hamilton’s equations for a particle with initial position a and no initial momentum, to find closed curve γ(t)=(x(t),p(t)), with x(0)=a and p(0)=0: In actuality, we are finding ˙x and ˙p. Starting with the former, ˙x: ∂x∂t=˙x=∂H∂p=∂∂p(p22+x22)=∂∂p(x22)+∂∂p(p22)=0+22p=p.

When can you Diagonalize a matrix?

The Jordan–Chevalley decomposition expresses an operator as the sum of its semisimple (i.e., diagonalizable) part and its nilpotent part. Hence, a matrix is diagonalizable if and only if its nilpotent part is zero.

What is diagonalization in linear algebra?

Diagonalizable A square matrix A is said to be diagonalizable if A is similar to a diagonal matrix, i.e. if A = PDP-1 where P is invertible and D is a diagonal matrix. Jiwen He, University of Houston. Math 2331, Linear Algebra.

What are Eigenstates?

Definition of eigenstate : a state of a quantized dynamic system (such as an atom, molecule, or crystal) in which one of the variables defining the state (such as energy or angular momentum) has a determinate fixed value.

Is the Hamiltonian matrix diagonal?

Such a hamiltonian is diagonal in the same basis that the following operators, are diagonal, where . This usually occurs when the Hamiltonian contains a term . However, when one adds a magnetic field, it introduces a term coupling to the magnetic moment of the system.

What is meant by linear harmonic oscillator?

The linear harmonic oscillator describes vibrations in molecules and their counterparts in solids, the phonons. Many more physical systems can, at least approximately, be described in terms of linear harmonic oscillator models.