by JUN BONITA
We examine a simple system in quantum mechanics. A particle is in a one dimensional infinite square well potential where the potential at a given length say L is zero and infinite elsewhere.
The solution to Schrodinger Equation for such a simple system consists of first knowing the initial wave function of the particle. That is, we first solve for wave function at time, t=0 which is given in details by:
This particular initial state is sketched below. We need to determine the initial wave function by finding the normalization constant A.
To determine A, we substitute the given wavefunction to the normalization condition and carry out the calculations as
Solution to the Schrodinger Equation,
The wave function for an infinite square well is then given as
From the wavefunction above, we must calculate the constant Cn,
At time ,the wave function reduces to
which we can write as
Then, cn can be calculated by applying inner product, that is,
And using the normalized initial wave functions
Recall that the integral can be solved using integral by parts,
This is easy to evaluate and obtain
Now we can answer the question as to the probability that a measurement of the energy will yield the value E1?
The energy levels of an infinite square well is given as
For the ground state, that is n=1 the energy is
This is the probability of getting the ground state energy is more than 98 %.
Expectation Values of the Hamiltionian Operator
The Hamiltonian of the quantum system is given by
where the potential energy function V(x) is equal to,
We first solve for the expectation value of the total energy.
The cross terms will vanish since the energy eigenstates are orthogonal to each other.
ABOUT THE AUTHOR:
JUN BONITA is finishing his M.S. Physics degree in the Mindanao State University-Iligan Institute of Technology (MSU-IIT), Iligan City, Philippines.