QM

• What is the main goal of Lecture 12: Properties of Wavefunctions?

  Normalise a continuous wavefunction
• Why is it necessary to make the state itself a continuous function in quantum mechanics?

  Because the spectrum of eigenvalues for position is continuous
• What is the Born rule for a continuous wavefunction?

  $P(x) =$$|\psi(x)|^2$
• How do you calculate the probability of finding a particle in a particular position interval?

  By multiplying the probability density by an interval of the variable
• What is the implication of a wavefunction being normalisable?

  The integral of P(x) over all space is finite-valued
• What is the normalisation condition for a wavefunction?

  $\langle\psi|\psi\rangle =$$1$
• How do you normalise a wavefunction that is not already normalised?

  By multiplying it by a normalisation coefficient: $1/\sqrt{N}$
• What is the implication of the continuity equation for a wavefunction?

  If a wavefunction is normalised at some time, then it is normalised at all times
• What is the probability current in the continuity equation?

  $J =$ -\frac{i\hbar}{2m}(\psi^\frac{\partial\psi}{\partial x} - \psi\frac{\partial\psi^}{\partial x})
• What is the condition for a wavefunction to be normalisable?

  $\langle\psi|\psi\rangle < \infty$
• What are the properties of a normalised wavefunction?

  • The integral of P(x) over all space is equal to 1
  • The wavefunction is a valid probabilistic representation of a physical state
  • The wavefunction can be used to calculate the probability of finding a particle in a particular position interval
• How do you calculate the probability of finding a particle in its ground state if it is in a superposition of states?

  By taking the square modulus of the projection of the ground state onto the superposition
• What is a wavefunction?

  A mathematical description of the quantum state of a system
• What is the difference between a discrete and continuous wavefunction?

  A discrete wavefunction has a countable number of eigenvalues, while a continuous wavefunction has an uncountable number of eigenvalues
• What is a superposition of states?

  A linear combination of two or more states
• What are the properties of a normalisable wavefunction?

  • The integral of P(x) over all space is finite-valued
  • The wavefunction is a valid probabilistic representation of a physical state
  • The wavefunction can be used to calculate the probability of finding a particle in a particular position interval
• What is the probability of finding a particle in its ground state given by?

  p = |⟨ψ0|Ψ⟩|2
• How is the probability of a particular state found in both discrete and continuous cases?

  By taking the square modulus of the projection
• What is the expression for the probability p in terms of the states ψ0 and ψ1?

  p = |⟨ψ0| 1/√2 ψ0 + 1/√2 ψ1⟩|2
• Why does only the first term contribute to the probability p?

  Because the states are orthogonal
• Does it matter if the eigenstates |ψ0⟩ and |ψ1⟩ are un-normalized in this case?

  No
• What additional information would be needed to find the probability of a particle within a certain range?

  Normalisation of the wavefunction w.r.t x
• What is the Triangle Inequality?

  |z1| + |z2| ≥ |z1 + z2|
• What is the Cauchy Inequality?

  2⟨z1|z1⟩⟨z2|z2⟩ ≥ ⟨z1|z2⟩ + ⟨z2|z1⟩
• What are the learning objectives of Lecture 13?

  Write differential operators, solve Schrödinger Equation, calculate reflection and transmission coefficients
• What are the properties of differential operators that correspond to physical observables?

  Hermitian, linear, and have real-valued eigenvalues
• What is the position operator in three dimensions?

  ˆx = x, ˆy = y, ˆz = z
• What is the momentum operator in three dimensions?

  ˆp = -iℏ∇
• What is the Hamiltonian operator for a non-relativistic particle?

  ˆH = ˆp2/2m + V(ˆx)
• What is the time-independent Schrödinger Equation for a free particle in one dimension?

  -ℏ²/2m d²ψ/dx² = Eψ
• What is the solution to the time-independent Schrödinger Equation for a free particle in one dimension?

  ψ(x) = Ae^{ikx}
• What happens to a free particle when a potential step is introduced?

  It scatters, with a reflected wave and a transmitted wave
• What can the probability density be interpreted as in certain cases?

  Average density of particles
• What is the conventional term for the state being described?

  Free particle
• What happens to a continuous beam of particles when a potential is introduced?

  It scatters
• What is the potential for $x < 0$?


  $0$
• What is the potential for $x \geq 0$?


  $U$
• What is the wavefunction for $x < 0$?


  $Ae^{ikx} +$$Be^{-ikx}$
• What is the energy of the particle in terms of $k$ and $m$?


  $\frac{\hbar^2k^2}{2m}$
• What conditions must the wavefunction satisfy at the boundary $x =$$0$?


  Continuity of $\psi(x)$ and $\frac{d\psi}{dx}$