Problem 1
Let’s consider one spatial dimension and time. Remember, we have defined the position and momentum operators in the class, xˆ and pˆ, respectively. The momentum operator is the derivative operator defined as :
(1)
where ℏ is the modified Planck’s constant. You now know that if ψ(x,t) is the wavefunction of the quantum system, then |ψ(x,t)|2 is the probability distribution function. For any operator Qˆ, the expectation of it is defined by the following equation :
Z
⟨Qˆ⟩ = dxψ(x,t)∗Qψˆ (x,t) (2)
1. Calculate the expectations of the position and momentum operators, ⟨xˆ⟩ and ⟨pˆ⟩.
2. Calculate the time derivative of the position expectation, d⟨xˆ⟩/dt.
3. We define the time derivative of the position expectation by vˆ. Show that ⟨p⟩ = m⟨v⟩.
4. The kinetic energy is defined as . Define the kinetic energy operator and calculate the expectation value of the kinetic energy operator ⟨Tˆ⟩.
5. Show that
(3)
This is known as Ehrenfest’s Theorem, which shows that the expectation values obey Newton’s Second law.
Problem 2 (15 marks)
In class, we derived the Probability conservation law in QM. The probability conservation tells us that the particle is conserved "locally" and is stable. Suppose you want to describe an "unstable" particle that spontaneously disintegrates with a lifetime of τ. In that case, the total probability of finding the particle somewhere should not be constant but decrease exponentially :
Z
P(t) ≡ dx|ψ(x,t)|2 = e−t/τ (4)
In our derivation, we used the fact that the potential energy V is real. This leads to the conservation of probability. What if we assign to V an imaginary part :
V = V0 − iΓ (5)
where V0 is the true potential energy and Γ is the positive real constant.
1. Calculate.
2. Solve for P(t) and find the lifetime of the particle in terms of Γ.
