The uncertainty principle

The uncertainty principle, ΔxΔp > ħ

Problem:

Use the uncertainty relation to find an estimate of the ground state energy of the harmonic oscillator.  The energy of the harmonic oscillator is E = p2/(2m) + ½mω2x2.

Solution:

Problem:

Consider the Hydrogen atom, i.e. an electron in the Coulomb field of a proton.  Use the uncertainty relation to find an estimate of the ground state energy of this system.

Solution:

Problem:

Electrons of kinetic energy 10 eV travel a distance of 2 km.  If the size of the initial wave packet is 10-9 m, estimate the size at the end of their travel.

Solution:

Problem:

Before the neutron was discovered one model assumed the atomic nucleus to be made of protons and electrons.  Show that the observation that the characteristic size of a nucleus is several times 10-15 m and that the average binding energy of a particle in the nucleus is less than 10 MeV makes this model inconsistent with basic principles of quantum mechanics. 

Solution:

Problem:

For a quantum mechanical point particle in a 1-dimensional harmonic potential U(x) = ½mω2x2
(a)  find the minimum (or “zero-point”) energy using the lower limit of Heisenberg's uncertainty principle,  ΔxΔp ≥ ħ/2.
(b)  For this zero-point energy of the particle, find the probability distribution by solving the time-independent Schroedinger equation for ψ(x).

Solution:

Problem:

Consider thermal neutrons in equilibrium at temperature T = 300 K.
(a)  Calculate its deBroglie wavelength.  State whether a beam of these neutrons could be diffracted by a crystal, and why?
(b)  Use Heisenberg's Uncertainty principle to estimate the kinetic energy (in MeV) of a nucleon bound within a nucleus of radius 10−15 m.


The uncertainty relation, ΔE Δt > ħ

Problem:

Assume that virtual π mesons are emitted and absorbed by a nucleus.  From this assumption, and the π meson mass, and the uncertainty principle, estimate the range of the nuclear potential r0

Solution:

Problem:

The transition rate of electrons from the first excited state of the hydrogen atom to the ground state is ~108/s.  What is the minimum range of energies of the resulting photons that are emitted?

Solution:

Problem:

Assume a particle with a mass of 105 MeV is the carrier of some interaction.  Estimate the range of this interaction.

Solution:


Generalized uncertainty principle

Problem:

Let A and B be two observables (Hermitian operators).  In any state of the system
ΔAΔB ≥ ½|<i[A,B]>|. 
(a)  Prove this generalized uncertainty principle.
[Hint: Let |ψ> be any state vector and let A1 = A - <A>I and B1 = B - <B>I. 
Let |Φ> = A1|ψ> + ixB1|ψ> with x an arbitrary real number.  Use <Φ|Φ>  ≥  0.]

Now consider a single particle in an eigenstate of L2 with wave function Ψ(r,t).
(b)  Calculate the commutators [sinφ, Lz] and [cosφ, Lz], where φ is the azimuthal angle.
(c)  Use these commutation relations and the result from part (a) to obtain uncertainty relations between sinφ, Lz and cosφ, Lz.
Note:  You can complete parts (b) and (c) without completing part (a).