Schrödinger (Time-Independent)
Fundamental equation of quantum mechanics.
Formula
Variables
How it works
The equation \(\hat{H}\psi = E\psi\) says that a stationary quantum state is an eigenfunction of the Hamiltonian, meaning the Hamiltonian acts on that wavefunction and gives back the same function scaled by a constant energy \(E\). It works because the Hamiltonian is the operator form of the total energy, so solving this equation is how we find allowed energy levels and the corresponding wavefunctions for systems whose potential does not change with time. In practice, it is the quantum version of finding the normal modes of a system, but with energy and wave behavior built into the mathematics.
When to use it
Use this equation whenever you need the allowed energies and stationary states of a time-independent quantum system, such as a particle in a box, a harmonic oscillator, or an electron in a fixed potential. It is the starting point for most bound-state problems in quantum mechanics.
Worked example, step by step
Consider a particle in a 1D infinite square well of width \(L=1.0\,\text{nm}\). For this system, the energy eigenvalues are \(E_n=\frac{n^2h^2}{8mL^2}\). Take an electron: \(m=9.11\times10^{-31}\,\text{kg}\), \(h=6.626\times10^{-34}\,\text{J·s}\), and choose \(n=2\). Substitute the values: \(E_2=\frac{(2)^2(6.626\times10^{-34})^2}{8(9.11\times10^{-31})(1.0\times10^{-9})^2}\) \(E_2=\frac{4(4.39\times10^{-67})}{8(9.11\times10^{-31})(1.0\times10^{-18})}\) \(E_2=\frac{1.76\times10^{-66}}{7.29\times10^{-48}}\) \(E_2\approx2.41\times10^{-19}\,\text{J}\) Convert to electronvolts: \(E_2=\frac{2.41\times10^{-19}}{1.60\times10^{-19}}\approx1.50\,\text{eV}\) Final answer: the second allowed energy level is \(E_2\approx2.41\times10^{-19}\,\text{J}\approx1.50\,\text{eV}\).
Common mistakes to avoid
- Treating \(\psi\) as a number instead of a function that must satisfy the operator equation
- Using the time-independent equation for systems with an explicitly time-dependent Hamiltonian
- Forgetting that \(E\) must be an allowed eigenvalue set by boundary conditions, not an arbitrary energy
Did You Know?
Schrödinger's equation is to quantum mechanics what Newton's laws are to everyday motion — the master equation.
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