a) Find a state of system at the time t

Section I, Quantum Physics

Postulates of Quantum Physics

1) A warm up. Suppose that a quantum system is described at the time t₀ by the state

|ψ(t₀)i = c₁|φ₁i + c₂|φ₂i ,

where c_i ∈ C and |φii are the eigenfunctions of Hamiltonian H corresponding to the eigenvalues E₁ and E₂.

a) Find a state of system at the time t.

b) Let B be an observable, [B, H] 6= 0, and b defines an eigenvalue of B corre- sponding to the eigenstate |ui. Find a probability P (b, ψ(t)), that measuring the quantity corresponding to B at the time t we get value b.

c) Show that P (b, ψ(t)) oscillates between extreme values with the frequency ν = |E₁− E₂|

h .

d) Explain the above result in view of the uncertainty principle.

2) Let us consider a quantum system governed by Hamiltonian H acting in the Hilbert space H. Suppose that H has a continuous spectrum. Assume that system is, at the time t = 0, described the function

|ψ(0)i = Z

Ω

dE c(E)|φ_Ei ,

where c(E) a function of energy determined by the set Ω = (E₀−∆E/2, E₀+

∆E/2) and |φ_Ei stands for the generalized eigenvector of H, i.e. ”H|φ_Ei = E|φEi”, however |φEi /∈ H.

a) Find a state |ψ(t)i of system at the time t.

b) Find a probability P (b, |ψ(t)i) that measuring the quantity corresponding to B the time t we get b; (B, b are defined at the point 1b)).

3) Gaussian wave packet. Let us consider a quantum system governed by the Hamiltonian

H = −~² 2m

d² dx² , 1

acting in the space L²(R). Let the (normalized) function ψ(x, 0) = 1

√2π Z ∞

−∞

g(k, 0)e^ikxdk , kψ(x, 0)k = 1 , describe the system at the time t = 0, where

√1

2πg(k, 0) =

√a

(2π)^3/4e^{−a24 (k−k0)2}. a) Show

ψ(x, 0) =

 2 πa²

1/4

e^ik0xe^−x2/a2. (Hint. Use R∞

−∞e^−ξ2dξ =√ π).

b) Auxiliary step. Show that if a state of system is described by f (x) = (_πb²2)^1/4e^−x2/b2 then

∆x =ph(X − hXi)²i = b

√2,

where X denotes position operator, h·i stands for mean value of an operator at a considered state.

c) Show that for the state ψ(x, 0) we have

∆x · ∆p = ~/2 ;

Hint. Calculate ∆x using the result of the point b). To obtain ∆p you can use the momentum representation of the momentum operator and apply it to the state g(k, 0). (k = ~p.)

Comment the result in view of the uncertainty principle.

d) Find the function ψ(x, t) corresponding to probability distribution at the time t. How does |ψ(x, t)|² behave as a function of time?

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