The Wavefunction

Chem 3240 · Lecture 3.2

Davit Potoyan

A Probabilistic World

  • The quantum world is fundamentally probabilistic.

  • Certainty is replaced by probabilities.

  • The wavefunction \(\psi\) is the central object.

Key question: what does \(\psi\) actually mean?

What is \(\psi\)?

  • Classically, a wave amplitude is a physical displacement (a guitar string).

  • The quantum \(\psi\) is generally complex.

  • \(\psi\) itself has no direct physical meaning.
  • We must extract real, measurable quantities from it.

The Born Rule

The absolute square is a probability density:

\[p(x) = \psi^{*}(x) \cdot \psi(x) = |\psi(x)|^2\]

  • \(p(x)\,dx\) = probability of finding the particle in \([x, x+dx]\).
  • In 3D: \(\quad p(x,y,z) = \psi(x,y,z)^{*} \cdot \psi(x,y,z)\)

Rules of Probability

  • Non-negative: \(\quad p(x) \geq 0\)

  • Normalized:

\[\int_{-\infty}^{\infty} p(x)\,dx = 1\]

  • Mean: \(\quad \mu = \langle x \rangle = \int_{-\infty}^{\infty} x\,p(x)\,dx\)
  • Variance: \(\quad \sigma^2 = \langle x^2 \rangle - \langle x \rangle^2\)

Normalization

A wavefunction must be normalizable to be a true probability distribution.

\[\int^{+\infty}_{-\infty} |\psi(x)|^2\, dx = 1\]

  • Guarantees the particle is somewhere in space.
  • Fix the constant in \(\psi = N\psi'\) by enforcing this condition.

Normalization Example

Normalize \(\psi(x) = C\,x\) on \([0,1]\):

\[C^2 \int_0^1 x^2\, dx = C^2 \cdot \frac{1}{3} = 1\]

\[C = \sqrt{3} \quad \Rightarrow \quad \psi(x) = \sqrt{3}\,x\]

Probability in a Region

Integrate the density over the region of interest:

\[p(a<x<b) = \int_a^b |\psi(x)|^2\, dx\]

For \(\psi = \sqrt{3}\,x\), so \(p(x) = 3x^2\):

\[P(0.3<x<0.6) = 0.6^3 - 0.3^3 = 0.189\]

Expectation Values

Weight values by their probability:

\[\langle x \rangle = \int x \cdot p(x)\, dx\]

Any function of \(x\):

\[\langle f \rangle = \int f(x) \cdot p(x)\, dx\]

Observables Are Operators

Momentum and energy are operators, not simple functions:

\[\langle A \rangle = \int \psi^{*}(x) \cdot \hat{A}\, \psi(x)\, dx\]

  • \(\hat{p} = -i\hbar\dfrac{d}{dx}\)
  • \(\hat{H} = -\dfrac{\hbar^2}{2m}\dfrac{d^2}{dx^2} + V(x)\)

Operators in Action

Particle in a box, \(\psi_n(x) = \sqrt{2}\sin(n\pi x)\):

\[\langle p \rangle = \int_0^1 \psi_n^*(-i\hbar)\psi_n'\, dx = 0\]

\[\langle p^2 \rangle = (n\pi\hbar)^2 \quad\Rightarrow\quad \langle E \rangle = \frac{(n\pi\hbar)^2}{2m}\]

Takeaway

The wavefunction is a probability amplitude: \(|\psi|^2\) is the probability density, normalization makes it certain the particle exists, and observables are expectation values of operators sandwiched in \(\psi^{*}\hat{A}\psi\).