A wave function describes the quantum state of an electron. A wave function is a mathematical model (or representation) of a given wave. Denoted by a greek letter \( \psi \). A square of wave function gives you the probability density of finding a particle at that point in space. \( \psi \) must be continuous and single valued everywhere.
In quantum mechanics, operators are the mathematical objects that represent measurable quantities, transforming the quantum state of a system to reveal its possible values. They are the fundamental link between the mathematical formalism of quantum theory and the physical world of experimental results.
Table1: Quantum mechanical operators corresponding to various observables(physical quantities)
| Observable | Quantum Mechanical Operator |
|---|---|
| Position (\(r\)) | \(\hat{r} = r\) |
| Momentum (\(p\)) | \(\hat{p} = -i\hbar \nabla = -i\hbar \dfrac{\partial}{\partial x}\) |
| Kinetic Energy (\(T\)) | \(\hat{T} = -\dfrac{\hbar^2}{2m} \nabla^2 = -\dfrac{\hbar^2}{2m} \dfrac{\partial^2}{\partial x^2}\) |
| Potential Energy (\(V\)) | \(\hat{V} = V\) |
| Total Energy, Hamiltonian (\(H\)) | \(\hat{H} = -\dfrac{\hbar^2}{2m} \nabla^2 + V\) |
In quantum mechanics, eigenfunctions and eigenvalues are fundamental concepts that describe the possible states and measurable values of a quantum system.
When a quantum operator acts on its eigenfunction, the result is the same function multiplied by a constant (the eigenvalue). This relationship is expressed by the eigenvalue equation:
\(\hat{A} \psi = a \psi\)
Where:
- \(\hat{A}\) is a quantum mechanical operator
- \(\psi\) is the eigenfunction of \(\hat{A}\)
- \(a\) is the eigenvalue corresponding to that eigenfunction
Adjust the parameters to see how the eigenfunctions and energy levels change: