QM Solved Problems #2

1.      A mono-energetic bundle of particles is moving along the x-axis to the right in a potential field V:

V = 0, x < 0;

V = V₀ > 0, x > 0;

      The energy E of the particles is greater than V₀.

a.       What are the solutions of the time-independent Schrödinger equation for x < 0 and x > 0?

b.      At x = 0, an incoming bundle from left, a reflected bundle and a transmitted bundle correspond to the boundary conditions. Calculate the reflection coefficient (the absolute value of the square of the amplitude of reflected and incoming bundles).

c.       If the energy of the particles is smaller than V₀, what are the solutions of the time-independent Schrödinger equation for x > 0 and the reflection coefficient?

Solution:

2.      A one dimensional rectangular potential field with energy E < 0, is given by:

V(x) = 0 for |x| > a;

V(x) = -V₀ < 0 for |x| ≤ a;

      Write the general solution of time independent Schrödinger equation for each area:

      x < -a, -a < x < a, x > a. Use the following definitions:

κ² = 2m|E|/ħ²;

q = 2m(V₀ - |E|)/ħ²

      Solution:

3.      Quantum mechanical states are vectors in a Hilbert space. A Hilbert space is a complex vector space with a Hermitian inner product. The properties of this inner product are that it is linear:

$<\phi |a{{\psi }_{1}}+b{{\psi }_{2}}>=a<\phi |{{\psi }_{1}}>+b<\phi |{{\psi }_{2}}>$

$<\phi |\psi >=<\psi |\phi >*$

$<\psi |\psi >\ge 0$

a.       Prove the Schwarz inequality: $|<\psi |\phi >{{|}^{2}}\le <\psi |\psi ><\phi |\phi >$. Hint: consider $<\phi +\lambda \psi |\phi +\lambda \psi >\ge 0$ and choose for λ the value which minimizes this expression.

b.      Prove the triangle inequality: $\sqrt{<\psi +\phi |\psi +\phi >}\le \sqrt{<\psi |\psi >}+\sqrt{<\phi |\phi >}$  a physical observable corresponds to a Hermitian operator.

c.       What is the definition of a Hermitian operator?

d.      Prove that the expectation value of a Hermitian operator is always real.

A Hermitian operator is called positive definite if for every ψ it holds that $<\psi |A|\psi >\ge 0$.

e.       A projection operator P is a Hermitian operator which satisfies P² = P. Show that the projection operator is always positive definite.

f.       Show that if A is a positive definite operator, $|<\psi |A|\phi >{{|}^{2}}\le <\psi |A|\psi ><\phi |A|\phi >$. Use the fact that you can expand in a basis of eigen-states and that the eigen-values of A are positive.

Solution:   

QM Solved Problems #1

1.      A particle moves along the x-axis in a potential field:

V(x) = 0, |x| < a

V(x) = ∞, |x| > a

(I.e. a box with closed, isolated walls at x = ± a).

a. Calculate the eigen-values of energy and define the corresponding normalized eigen-functions.

b. At t = 0, the particle is in interval |x| < c, (c < a). Inside the interval, all positions of the particle have the same chances. Assume that the corresponding wave function is real and positive. Please find the chance at energy measurement of finding the particle at:

i.   the ground state;

ii.  the first excited state;         

c. Assume that at the energy measurement, the ground state was found and then, we suddenly remove the walls of the box to x = ± b (b > a). What is then the probability of finding the particle at:

            i.   the ground state?

            ii.  the first excited state?

           

            Solution: