Calculating eigenvectors/values from Hamiltonian

In summary, the conversation discusses the process of constructing a Hamiltonian by using a 2nd order numerical derivative stencil and adding the potential, and then solving for the eigenvalues and corresponding eigenvectors using the "eig" function in MATLAB. There is a concern about the behavior observed in the probability density, and the potential may need to be referenced to the lowest point to avoid skewing the calculation. The suggestion is to change the units and make sure the matrix eigenvalue equation has the factors of 1/(\Delta x)^2 in the matrix.
  • #1
fenny
1
0

Homework Statement


I've constructed a 3D grid of n points in each direction (x, y, z; cube) and calculated the potential at each point.
For reference, the potential roughly looks like the harmonic oscillator: V≈r2+V0, referenced from the center of the cube.
I'm then constructing the Hamiltonian by using a 2nd order numerical derivative stencil in each direction representing the Laplacian seen in the TISE, multiplying each stencil by the factor ## \frac {ħ} {2m} ##, and adding the potential.

Homework Equations


## H = \frac {-ħ^2} {2m} ∇^2 + V ##

## ∇^2 = \frac {∂^2} {∂x^2} +\frac {∂^2} {∂y^2} +\frac {∂^2} {∂z^2} ##

The Attempt at a Solution


My approach was to simply solve for the eigenvalues and corresponding eigenvectors of H (currently using "eig" function in MATLAB). However, I generally observe peaks at the corners or edges when verifying the probability density observed (## Ψ* Ψ ##) and approximately zero elsewhere in the cube.
I've gone through my equations and verified the units to be correct a few times now, and I'm wondering if something is being missed on my end or if something is fundamentally flawed with this approach. Eventually I would also like to solve the problem using the variational method as well to compare the answers (and double check my calculation/method).
For a grid with n points along each axis, n3 eigenvectors/values are obtained due to construction of a n3 × n3 matrix representing H. I initially examined the lowest energy states primarily but am now sampling along the different energies and still observe the same behavior.
I'm also wondering if my potential (V) should be referenced to the lowest point (near the center), essentially removing the constant term. My fear was that the increase along the H matrix diagonal could be skewing a calculation. As it is, my potential terms (added to the primary diagonal) appear to be ~10-10 while my stencil terms considering the neighboring points (adjacent to Ψi,j,k) are approximately ~10-15 due to small reduced Planck constant.
 
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  • #2
You need to do some kind of change of units that makes the terms closer to unity, for example ##\hbar = 1, m=1##. Also, make sure that the matrix eigenvalue equation ##H\psi = E\psi## has the factors of ##1/(\Delta x)^2## in the matrix, not in the RHS vector (the nonzero matrix elements will then become relatively large numbers). That way you should get a decent solution.
 

1. What is a Hamiltonian in physics?

The Hamiltonian is a mathematical operator in quantum mechanics that describes the total energy of a system. It is used to calculate the time evolution of a physical system and can provide information about the energy levels and properties of the system.

2. How are eigenvectors and eigenvalues related to the Hamiltonian?

Eigenvectors and eigenvalues are important concepts in linear algebra that are used to solve for the energy levels of a system described by the Hamiltonian. The eigenvectors are the basis states of the system, while the eigenvalues represent the corresponding energy levels.

3. What is the process for calculating eigenvectors and eigenvalues from the Hamiltonian?

The process for calculating eigenvectors and eigenvalues from the Hamiltonian involves solving the eigenvalue equation, which is a mathematical equation that relates the eigenvectors and eigenvalues of a linear operator. This is typically done using techniques such as diagonalization or matrix diagonalization.

4. What is the significance of the eigenvectors and eigenvalues in quantum mechanics?

Eigenvectors and eigenvalues are essential in quantum mechanics as they provide information about the energy levels and properties of a system described by the Hamiltonian. They are used to determine the probability of a particle being in a certain energy state and can also be used to study the behavior of quantum systems.

5. Are there any practical applications of calculating eigenvectors and eigenvalues from the Hamiltonian?

Yes, there are several practical applications of calculating eigenvectors and eigenvalues from the Hamiltonian. For example, in quantum chemistry, these calculations are used to study the electronic structure of molecules and predict their chemical properties. They are also used in quantum computing, where eigenvectors and eigenvalues are used to perform operations on quantum bits (qubits). Additionally, these calculations are important in fields such as materials science and engineering, where they can help predict the behavior of materials at the atomic level.

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