Problem Class Week 14 Problems
Updated: 29 Jan 2026
These are the problems that we will solve during the problem class in week 14. You are not expected to attempt these questions in advance of the problem class. We will work through them in sequence, going through the solutions after everybody has time to attempt the questions.
I have included additional questions for you to attempt / discuss if you complete the questions before we go over the solution. If you also complete these, please work through Problem Sheet: Week 14
Problems ¶ What is the action of the operator X ^ 2 \hat{X}^2 X ^ 2 ∣ x ⟩ \ket{x} ∣ x ⟩
What is the action of the operator X ^ 2 \hat{X}^2 X ^ 2 ∣ ψ ⟩ = ∫ − ∞ ∞ ψ ( x ) ∣ x ⟩ d x \ket{\psi} = \infint \psi(x) \ket{x} dx ∣ ψ ⟩ = ∫ − ∞ ∞ ψ ( x ) ∣ x ⟩ d x
Write down the general relationship between an operator A ^ \hat{A} A ^ ∣ ψ ⟩ \ket{\psi} ∣ ψ ⟩ A ^ w \op{A} A ^ w
Use parts 2. and 3. to write down the operator X ^ w 2 \op{X}^2 X ^ w 2
What is the eigenvalue equation for the operator X ^ 2 \hat{X}^2 X ^ 2
Can you spot what the eigenstates of X ^ 2 \hat{X}^2 X ^ 2
Is X ^ 2 \hat{X}^2 X ^ 2 degenerate operator? (that is, are there multiple eigenstates which have the same eigenvalue?) If yes, how many states share an eigenvalue?
What is the physical significance of an operator being degenerate or not?
Write down the operator P ^ w \op{P} P ^ w P ^ \hat{P} P ^
Calculate the commutator [ X ^ w 2 , P ^ w ] ψ ( x ) [\op{X}^2, \op{P}]\psi(x) [ X ^ w 2 , P ^ w ] ψ ( x ) ψ ( x ) \psi(x) ψ ( x )
Use part 2. to write down the commutator [ X ^ 2 , P ^ ] [\hat{X}^2, \hat{P}] [ X ^ 2 , P ^ ]
What is the physical significance of your answer to part 3?
Calculate [ X ^ w 3 , P ^ w ] ψ ( x ) [\op{X}^3, \op{P}]\psi(x) [ X ^ w 3 , P ^ w ] ψ ( x ) [ X ^ 3 , P ^ ] [\hat{X}^3, \hat{P}] [ X ^ 3 , P ^ ]
Use part 5. to make a guess for what the commutator [ X ^ n , P ^ ] [\hat{X}^n,\hat{P}] [ X ^ n , P ^ ] n n n
Mathematically show that your guess from part 6 is correct.
What is the action of the operator P ^ \hat{P} P ^ ∣ ψ ⟩ = ∫ − ∞ ∞ ψ ( x ) ∣ x ⟩ d x \ket{\psi} = \infint \psi(x) \ket{x} dx ∣ ψ ⟩ = ∫ − ∞ ∞ ψ ( x ) ∣ x ⟩ d x
What is the action of the operator P ^ 2 \hat{P}^2 P ^ 2 ∣ ψ ⟩ = ∫ − ∞ ∞ ψ ( x ) ∣ x ⟩ d x \ket{\psi} = \infint \psi(x) \ket{x} dx ∣ ψ ⟩ = ∫ − ∞ ∞ ψ ( x ) ∣ x ⟩ d x
Use part 2. to write down P ^ w 2 \op{P}^2 P ^ w 2
Calculate [ X ^ w , P ^ w 2 ] ψ ( x ) [\op{X}, \op{P}^2]\psi(x) [ X ^ w , P ^ w 2 ] ψ ( x ) [ X ^ , P ^ 2 ] [\hat{X}, \hat{P}^2] [ X ^ , P ^ 2 ]
Use part 5. to make a guess for what the commutator [ X ^ , P ^ n ] [\hat{X},\hat{P}^n] [ X ^ , P ^ n ] n n n
Mathematically show that your guess from part 5 is correct.
Consider the particle from last week’s problem class, (initially) confined to the region x ≥ 0 x \geq 0 x ≥ 0
∣ ψ i n ⟩ = 2 L ∫ 0 ∞ e − x / L ∣ x ⟩ d x , \ket{\psi_\mathrm{in}} = \sqrt{\frac{2}{L}}\int_0^\infty e^{-x/L} \ket{x} dx, ∣ ψ in ⟩ = L 2 ∫ 0 ∞ e − x / L ∣ x ⟩ d x , where L > 0 L > 0 L > 0 there are no forces acting on the particle so that it is a free particle .
Write down the equation from the notes, which gives the (generic) quantum state at time t t t ∣ ψ ( t ) ⟩ \ket{\psi(t)} ∣ ψ ( t ) ⟩ t = 0 t = 0 t = 0 ψ ~ i n ( p ) \tilde{\psi}_\mathrm{in}(p) ψ ~ in ( p )
Look up the momentum wavefunction of Equation t = 0 t = 0 t = 0 t t t ∣ ψ ( t ) ⟩ \ket{\psi(t)} ∣ ψ ( t ) ⟩
Write down (an integral) which gives the spatial wavefunction ψ ( x , t ) \psi(x,t) ψ ( x , t )
Think about how you might numerically integrate this, to see the evolution.
Solutions ¶