Introduction

Classical view

• Matter is described with particles
• Newton’s equation:

\[ \vec{F} = m \, \vec{a} = \frac{\partial^2 r_i}{{\partial t}^2} \]

• Forces act on point masses
• The force is the gradient of the potential energy:

\[ \vec{F} = - \nabla V \]

• Conservation of energy \(E = E_{kin} + E_{pot} = T + V\)

Classical view

• Light is described by waves
• Maxwell’s equations

\[ \left\{\begin{aligned} \nabla \cdot \vec{E} &= \frac{\rho}{\epsilon_0}\\ \nabla \cdot \vec{B} &= 0\\ \nabla \times \vec{E} &= - \frac{\partial \vec{B}}{\partial t}\\ c^2 \, \nabla \times \vec{B} &= \frac{\vec{J}}{\epsilon_0} + \frac{\partial \vec{E}}{\partial t}\\ \end{aligned}\right . \]

Classical view

• Light is described by waves
• Maxwell’s equations
• If there are no charges or currents then \(\rho = 0\) and \(\vec{J} = 0\) then:

\[ \left\{\begin{aligned} \nabla^2 \vec{E} - \frac{1}{c^2}\frac{\partial^2 \vec{E}}{\partial t^2} &= 0 \\ \nabla^2 \vec{B} - \frac{1}{c^2}\frac{\partial^2 \vec{B}}{\partial t^2} &= 0 \\ \end{aligned}\right . \]

Classical view

Light and matter are treated different

• Light has wave-like behavior
• Matter exists of particles

Problems:

• Hydrogen atom: Electron should fall on nuclues
• Specific energy bands of atomic spectra?
• Electrons can tunnel through potential energy barriers

The Schrödinger equation

\[ i \hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \Psi + V \Psi \]

where

• Complex wave function: \(\Psi \rightarrow \Psi(x, y, z, t)\)
• Laplacian \(\nabla^2 = \nabla\cdot\nabla = \frac{\partial^2}{\partial^2 x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial}{\partial z^2}\)
• potential energy: \(V \rightarrow V(x,y,z,t)\)
• \(\hbar = \frac{h}{2\pi} = 1.055 \times 10^{-34}\) J s

The Schrödinger equation

We will first consider 1D problems:

\[ i \hbar \frac{\partial \Psi(x,t)}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2 \Psi(x,t)}{\partial x^2} + V(x,t) \Psi(x,t) \]

Probabilistic view

\[ P(x \in [a,b]) = \int_a^b |\Psi(x, t)|^2 dx \]

Probabilistic view: Measerement problem

Copenhagen interpretation:

• Before measurement: probability according to \(|\Psi(x, t)|^2\)
• Measurement: Wave function collapses to a single state \(\longrightarrow\) \(\delta\)-function
• After measurement: \(\delta\)-function spreads out again over time.

Double slit experiments:

Typical thought-experiment

• What happens if electron “particles” are fired through a double slit?
• What happens if light at low intensity (single photons) is used?

Probability and Expectation values

Probability density function \(\rho(x)\):

\[ \begin{aligned} \textrm{Expectation value of }x \qquad & \langle x \rangle = \int_{-\infty}^\infty x \,\rho(x)\,dx\\ \textrm{Expectation value of }f(x) \qquad & \langle f(x) \rangle = \int_{-\infty}^\infty f(x) \,\rho(x)\,dx\\ \end{aligned} \]

Probability and Expectation values

Probability density function \(\rho(x)\):

\[ \begin{aligned} \textrm{Expectation value of }x \qquad & \langle x \rangle = \int_{-\infty}^\infty x \,\rho(x)\,dx\\ \textrm{Expectation value of }f(x) \qquad & \langle f(x) \rangle = \int_{-\infty}^\infty f(x) \,\rho(x)\,dx\\ \textrm{Variance }\sigma^2 \qquad & \langle (\Delta x)^2 \rangle = \int_{-\infty}^\infty (x - \langle x \rangle)^2 \,\rho(x)\,dx\\ \qquad & \qquad\quad = \langle x^2 \rangle - \langle x \rangle^2\\ \textrm{Standard deviation } \qquad & \sigma = \sqrt{\langle x^2 \rangle - \langle x \rangle^2}\\ \end{aligned} \]

Normalization of the wave function

• \(|\Psi(x, t)|^2\) is like the probability density \(\rho(x)\)
• The total probability to find a particle somewhere must be one:

\[ \int_{-\infty}^\infty |\Psi(x, t)|^2 dx = 1 \]

Normalization of the wave function

If \(\Psi(x, t)\) is normalized at \(t = 0\) then it is always normalized.

Expectation values

What are the particle’s:

• position \(x\)?
• velocity \(v\) ? or
• momentum \(p = mv\) ?

Position and Momentum operators

Expectation values are calculated as

Schrodinger equation with operators

\[ i \hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x} \Psi + V \Psi \]

We have the operators:

\[ \textrm{Position operator} \qquad \hat{x} = x \] \[ \textrm{Momentum operator} \qquad \hat{p} = -i\hbar\frac{\partial }{\partial x} \]

Correspondence principle

• Large systems: Quantum mechanics \(\longrightarrow\) classical physics
• Ehrenfest’s theorem:

\[ m \frac{d}{dt}\langle x \rangle, \qquad \frac{d}{dt}\langle p \rangle = - \langle\frac{\partial V(x)}{\partial x}\rangle \]

Uncertainty relation: position vs. momentum

• de Broglie relation

\[ p = \frac{h}{\lambda} = \frac{2\pi\hbar}{\lambda} \quad (= \hbar k) \]

• Think about a Gaussian wave pulse in Fourier analysis
  • Sharp pulses in space are spread out in (momentum) k-space
  • Sharp pulses in k-space are spread out in space
• Uncertainty of position vs. momentum

\[ \sigma_x \sigma_p \ge \frac{\hbar}{2} \]

Summary

• Quantum mechanics is governed by the Schrodinger equation

\[ i \hbar \frac{\partial \Psi}{\partial t} = -\frac{\hbar^2}{2m} \nabla^2 \Psi + V \Psi \]