Summary of what we know

• States \(|\Psi\rangle\) can be represented by the wave function:
  • \(\Psi(x,t) = \langle x | \Psi(t) \rangle\)
  • this is similar to a vector in vector component notation

Observables, operators and collapse

• We can measure observables:
  • position and momentum of a particle,
  • energy of a particle in a potential,
  • excitation-level of an electron in an atom
  • spin of an electron
  • …

Infinite well

infinite well: Observable position

infinite well: energies

infinite well: Observable energy

wavepacket incident on barrier

wavepacket: Observable position

Observables, operators and collapse

• State of a quantum system: \(| \Psi \rangle\)
• Wave function represents state: \(\langle x | \Psi(t) \rangle \longrightarrow \Psi(x, t)\)
• Observable is something we can measure (a real number)
• Observable \(Q\) corresponds to an Hermitian operator \(\hat{Q}\)
• Measuring NOT same as applying operator \(\hat{Q} | \Psi \rangle\)

Uncertainty principle

• Heisenberg uncertainty principle

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

• Commutator is nonzero:

\[ [\hat{x}, \hat{p}] = \hat{x}\hat{p} - \hat{p}\hat{x} = i\hbar \]

• Can’t measure position and momentum at the same time
• Measuring position destroys the momentum measurement

Generalized Uncertainty principle

• General uncertainty principle is related to the commutator

\[ \boxed{\sigma_A^2\sigma_B^2 \geq \left( \frac{1}{2i} \left\langle \left[ \hat{A}, \hat{B} \right] \right\rangle\right)^2} \]

• Number between brackets is real but can be negative
• We need the square at the right-hand-side
• Commutating operators \(\longrightarrow\) no restriction on \(\sigma_A\), \(\sigma_B\)

How to proof this?

Example uncertainty principle

• General uncertainty principle for position/momentum
• The commutator for \(\hat{x}\) and \(\hat{p}\):

\[ [\hat{x}, \hat{p}] = i\hbar \]

Fill in in general uncertainty formula:

\[ \begin{aligned} \sigma_A^2\sigma_B^2 &\geq \left( \frac{1}{2i} \left\langle \left[ \hat{A}, \hat{B} \right] \right\rangle\right)^2\\ \Rightarrow\quad\sigma_x^2\sigma_p^2 &\geq \left( \frac{1}{2i} \langle \left[ \hat{x}, \hat{p} \right] \rangle\right)^2 = \left( \frac{1}{2i} \langle i\hbar \rangle\right)^2 = \frac{\hbar^2}{4}\\ \Rightarrow\quad\sigma_x\sigma_p &\geq \frac{\hbar}{2}\\ \end{aligned} \]

\(\longrightarrow\) Heisenberg uncertainty principle

Commutators and uncertainty

\[ \boxed{\sigma_A^2\sigma_B^2 \geq \left( \frac{1}{2i} \left\langle \left[ \hat{A}, \hat{B} \right] \right\rangle\right)^2} \]

• Compatible observables: Commutating observables \(\quad [\hat{A}, \hat{B}] = 0\)
  • Measurements independent, order doesn’t matter
  • No restriction on the common uncertainty of the measurement
  • A common basis of eigenstates can be found