If quantum cosmologists can reconcile quantum mechanics with Einstein's Theory of General Relativity, we may be in for a wild ride, as scientists will finally have a single theory with the potential to describe all aspects of time and space in the universe. In short, we will be on our way to knowing the unknowable. Quite exciting, to say the least. At the heart of quantum theory lies such ideas as the particle nature of waves, the wave-like properties of particles, the quantized energy levels of atoms, and the Heisenberg Uncertainty Principle. It is from these fundamental assumptions about the natural world that we may be able to unlock its secrets through the development of a unified field theory. A good course in quantum physics will take students through the following areas:

  • The wave function
  • Time-independent Schrodinger equation
  • Formalism
  • Quantum mechanics in three dimensions
  • Identical particles
  • Time-independent perturbation theory
  • The variational principle
  • The WKB approximation
  • Time-dependent perturbation theory
  • The adiabatic approximation
  • Scattering

There are plenty of books on quantum physics to be found on Google and Amazon.com. There is also a nice collection of books and journals devoted to this area which students can find at the Springer website.

 

The variational principle

If a quantum system has a Hamiltonian independent of time  and one chooses a normalized “well behaved” function  (i.e. a continuous function for which the first and second derivatives exist) then

where  is the lowest energy eigenvalue of the Hamiltonian (bound energy).

In other words, if we chose a random test function  the bounded energy will be minimum of the expression

Question 1

Consider a one-dimensional system of a particle with a potential energy:

Make a graph of V vs X. Use the test function  to determine the energy of the ground level. Note that there is no variational parameter.  Note that is given by the equation (why?):

Solution:

If you consider a test function  having no variational parameter (that is having no constant coefficients inside) then the eigenvalue of the bound level can be approximated by

 

Since

We can write:

since we have:

Question 2

Consider you have a test function:

where f1 and f2 are real functions. Find the energy of the bound level in terms of the integrals

For the special case when:

 

find also the rapport of the coefficients:  

Solution:

If you have a test function of the type

Then the energy of the bound level is given by (like above):

 

Which for:

 

gives the rapport of coefficients:

 

To have a solution of the above system of two equations one needs to have the characteristic determinant = 0. In other words the condition for existence of E states is:

or

 

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