Table of Contents
Overview#
The wave function \(\Psi\) is the fundamental mathematical description of quantum systems. It contains all information about a particle’s quantum state.
Definition#
The wave function \(\Psi(x, t)\) is a complex-valued function:
$$ \Psi(x, t) = A e^{i(kx - \omega t)} $$Where:
- \(k = \frac{2\pi}{\lambda}\): Wave number
- \(\omega = 2\pi f\): Angular frequency
- \(A\): Amplitude
Physical Interpretation#
Born’s Probability Interpretation#
The probability of finding a particle between \(x\) and \(x + dx\):
$$ P(x) dx = |\Psi(x)|^2 dx = \Psi^* \Psi \, dx $$Normalization Condition#
Total probability must equal 1:
$$ \int_{-\infty}^{\infty} |\Psi(x)|^2 dx = 1 $$The Schrödinger Equation#
Time-Dependent#
$$ i\hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \Psi}{\partial x^2} + V(x)\Psi $$Or in operator form:
$$ i\hbar \frac{\partial \Psi}{\partial t} = \hat{H}\Psi $$Time-Independent#
For stationary states \(\Psi(x,t) = \psi(x)e^{-iEt/\hbar}\):
$$ -\frac{\hbar^2}{2m}\frac{d^2\psi}{dx^2} + V(x)\psi = E\psi $$Important Examples#
Free Particle#
\(V(x) = 0\):
$$ \Psi(x, t) = Ae^{i(kx - \omega t)} $$Energy relation:
$$ E = \frac{\hbar^2 k^2}{2m} = \frac{p^2}{2m} $$Infinite Square Well#
\(V = 0\) for \(0 < x < L\), \(V = \infty\) otherwise:
$$ \psi_n(x) = \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi x}{L}\right) $$Energy levels:
$$ E_n = \frac{n^2 \pi^2 \hbar^2}{2mL^2} $$Harmonic Oscillator#
\(V(x) = \frac{1}{2}m\omega^2 x^2\):
$$ \psi_n(x) = \left(\frac{m\omega}{\pi\hbar}\right)^{1/4} \frac{1}{\sqrt{2^n n!}} H_n(\xi) e^{-\xi^2/2} $$Where \(\xi = \sqrt{\frac{m\omega}{\hbar}}x\) and \(H_n\) are Hermite polynomials.
$$ E_n = \hbar\omega\left(n + \frac{1}{2}\right) $$Properties of Wave Functions#
Superposition#
If \(\Psi_1\) and \(\Psi_2\) are solutions, so is:
$$ \Psi = c_1\Psi_1 + c_2\Psi_2 $$Expectation Values#
Position:
$$ \langle x \rangle = \int_{-\infty}^{\infty} \Psi^* x \Psi \, dx $$Momentum:
$$ \langle p \rangle = \int_{-\infty}^{\infty} \Psi^* \left(-i\hbar\frac{\partial}{\partial x}\right) \Psi \, dx $$Uncertainty#
$$ \Delta x = \sqrt{\langle x^2 \rangle - \langle x \rangle^2} $$$$ \Delta p = \sqrt{\langle p^2 \rangle - \langle p \rangle^2} $$Heisenberg uncertainty principle:
$$ \Delta x \cdot \Delta p \geq \frac{\hbar}{2} $$Wave Function Collapse#
Upon measurement:
- Wave function “collapses” to eigenstate
- Probability becomes certainty
- Copenhagen interpretation
Dirac Notation#
| Notation | Meaning |
|---|---|
| \(\ket{\psi}\) | State vector (ket) |
| \(\bra{\phi}\) | Dual vector (bra) |
| \(\braket{\phi|\psi}\) | Inner product |
| \(\ket{\psi}\bra{\phi}\) | Outer product |