Postulates(1)

Postulates of Quantum Mechanics

Problem:

A particle is described by the wavefunction y(x)=(2L)^-1/2 for |x| £ L, and y(x)=0 for |x| > L.

(a) What is the probability P(p) of finding the particle with momentum p?

(b) Sketch the probability P(p) and discuss it in relation to the uncertainty principle.

Solution:

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Problem:

(a) The wavefunction for a spinless particle moving in a central potential V(r)=V(r) is y=A sinq f(r), where A is a constant, and What is the probability that a measurement of the angular momentum quantum number l would yield 0? 1?

(b) In the same potential suppose that y=Bsinq sinf f(r), where B is a constant, and What values and with what probabilities would be obtained for measurements of the total angular momentum and the z-component of the angular momentum?

Solution:

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Problem:

A particle of mass m is contained in a one-dimensional impenetrable box extending from x=-L/2 to x=L/2. The particle is in its ground state. The walls of the box are now moved out, symmetrically an instantaneously, to form a box extending from x=-L to x=L.

(a) Calculate the probability, that after this sudden expansion the particle will be in the ground state.
(b) What is the probability that it will be found in the first excited state?
(c) What is the expectation value of the energy after the rapid expansion? Answer without calculation.

Solution:

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Problem:

Two infinite potential wells are extending from x=-a to x= 0 and from x= 0 to x=a, respectively. A particle is in its ground state in the left well. At t=0 the barrier at x=0 is removed. What is the probability of finding the particle in the first excited state of the new well?

Solution:

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Problem:
An atom with L=0 and S=½ is initially in its ground state with S_z=-½. A field H₁i is turned on at t=0 so that the effective Hamiltonian may be taken to be H=H₀-(e/m_ec)S_xH₁q(t), where H₀ is the atomic Hamiltonian with H₁=0. Thus both S_z=½ and S_z=-½ states have the same eigenvalue of H₀ in the ground state.
(a) What is the probability P, to lowest order in H₁, that the atom remains in the state |S=½,S_z=-½> at time t? Here

, e is the magnitude of the electron charge, m_e is the electron mass, and c is the velocity of light in the vacuum.
(b) Under what conditions on t will P=0?

Solution:
Concepts, principles, relations that apply to the problem:
Two level systems, the postulates of quantum mechanics.

Why do they apply?
Assume that the energy of the twofold degenerate ground state is much lower than the energy of any excited state, and approximate the system by a two-level system. Then the eigenstates of H₀ are known and form a basis for the two-dimensional state space of the system. We can diagonalize the matrix of H and find the eigenvalues of H and the eigenfunctions of H as linear combinations of eigenfunctions of H₀. The initial state is a linear combination of these eigenstates, and we therefore know how it evolves in time. The postulates of quantum mechanics tell us that P=|<y (0)|y (t)>|².

How do they apply?
Denote the two levels by |+> and |->. H₀|+>=E₀|+>, H₀|->=E₀|->, the system is degenerate. {|+>,|->} are the eigenstates of S_z. .

At t=0 the system is in the state |->, but the Hamiltonian is H₀+W=H, with . In the {|+>,|->} the matrix of W is . The matrix of H is .

The eigenvalues of H are , and the corresponding eigenfunctions are .

Details of the calculations:

In this problem . .

. .

If then P_-(t)=0.

If with n an odd integer, then P_-(t)=0.