Introduction: Two Views of the Same State
A quantum state \(|\psi\rangle\) can be described in the position representation (wavefunction \(\psi(x)\)) or the momentum representation (\(\tilde{\psi}(p)\)). They contain the same information, related by Fourier transform.
Position Representation
In position representation:
- State: \(\psi(x) = \langle x|\psi\rangle\)
- Position operator: \(\hat{x}\psi(x) = x\psi(x)\) (just multiplication)
- Momentum operator: \(\hat{p}\psi(x) = -i\hbar\frac{d}{dx}\psi(x)\) (differentiation)
Momentum Representation
In momentum representation:
- State: \(\tilde{\psi}(p) = \langle p|\psi\rangle\)
- Momentum operator: \(\hat{p}\tilde{\psi}(p) = p\tilde{\psi}(p)\) (just multiplication)
- Position operator: \(\hat{x}\tilde{\psi}(p) = i\hbar\frac{d}{dp}\tilde{\psi}(p)\) (differentiation)
The Fourier Transform Connection
\[\tilde{\psi}(p) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} e^{-ipx/\hbar} \psi(x) \, dx\] \[\psi(x) = \frac{1}{\sqrt{2\pi\hbar}} \int_{-\infty}^{\infty} e^{ipx/\hbar} \tilde{\psi}(p) \, dp\]Worked Examples
Example 1: Gaussian Wavepacket
A Gaussian in position space:
\[\psi(x) = \left(\frac{1}{\pi a^2}\right)^{1/4} e^{-x^2/2a^2}\]Its Fourier transform is also Gaussian:
\[\tilde{\psi}(p) = \left(\frac{a^2}{\pi\hbar^2}\right)^{1/4} e^{-a^2p^2/2\hbar^2}\]Width in x times width in p gives a constant (the uncertainty relation).
Example 2: Plane Wave
A momentum eigenstate \(|p_0\rangle\) in position representation:
\[\langle x|p_0\rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{ip_0 x/\hbar}\]A plane wave with wavelength \(\lambda = 2\pi\hbar/p_0\).
Example 3: Position Eigenstate in Momentum Space
\[\langle p|x_0\rangle = \frac{1}{\sqrt{2\pi\hbar}} e^{-ipx_0/\hbar}\]Also a plane wave—the dual relationship is symmetric.
The Quantum Connection
The position and momentum representations are "conjugate" to each other. A state localized in position is spread in momentum, and vice versa. This is the mathematical content of the Heisenberg uncertainty principle:
\[\Delta x \cdot \Delta p \geq \frac{\hbar}{2}\]You're now ready to study operators and eigenvalues in depth.