Chemistry 3730 Fall 1997 Test 1 Solutions

1. We are asked to calculate the energy of a photon given its wavelength:

   

2. Heisenberg's uncertainty principle states that . In this paper, it is claimed that while can be as small as 0.01m/s (1% of 1m/s). For protons, this gives us a of

   

   Thus, for this experiment, it is claimed that the product can be as small as . On the other hand, . The claimed accuracy of the experiment is much greater than is permitted by the uncertainty principle. The author of the article has either made a mistake or has found a way around a fundamental limitation of quantum mechanics. The former is considerably more likely.

3. First, let's define the wavefunction:
   

   > psi := (n,L,x) -> sqrt(2/L)*sin(n*Pi*x/L);

   

   
   

   The ground state of a particle in a box is the n=1 state. The probability is
   

   > int(psi(1,L,x)^2,x=3*L/5..2*L/3);

   

   > evalf(");

   

   
   

4. The expectation value is the inner product :
   

   > int(psi(3,L,x)*x^15*psi(3,L,x),x=0..L);

   

   > evalf(");

   

   
   

5. The commutator is

   

   The last term is zero because after applying the commutator simplification rules, it can only produce terms involving . We apply the commutator rule for products to the other two terms:

   

   If the commutator of two observables is zero, they can be measured simultaneously without one measurement interfering with the other. We say that two such observables are compatible.

6. If the wavefunction is normalized, . In this case,

   

   Normalization is required for to be interpretable as a probability density.

7. 1. For n=1, or . For n=2, . For n=3, . The general pattern is (an integer number of half-wavelengths fits in the box) or, to put it the other way around,

      

   2. 

      

   3. For a particle in a box, all of the energy is kinetic and ,

      

      Unfortunately, this kind of reasoning only works for a very small number of quantum mechanical problems.