Problem 8.1 - Variational Principle Theory ⇢ Gaussian Trial Function: Intro to Quantum Mechanics

Introduction

This tutorial explores the variational principle in quantum mechanics using a Gaussian trial function to estimate the ground state energy for two specific potentials: the linear potential ( V(x) = \alpha |x| ) and the quartic potential ( V(x) = \alpha x^4 ). By following these steps, you'll gain a deeper understanding of how trial functions can be utilized in quantum mechanics.

Step 1: Understand the Variational Principle

• The variational principle states that the ground state energy of a quantum system is always less than or equal to the expectation value of the Hamiltonian calculated with any trial wave function.
• This principle allows for estimating the ground state energy by optimizing parameters in the trial wave function.

Step 2: Define the Gaussian Trial Function

• A Gaussian trial function can be defined as:

  [ \psi(x) = A e^{-\alpha x^2} ]

• Here, ( A ) is a normalization constant and ( \alpha ) is a parameter that will be adjusted to minimize the energy.

Step 3: Normalize the Wave Function

• To find the normalization constant ( A ), ensure that the wave function satisfies the normalization condition:

  [ \int_{-\infty}^{\infty} |\psi(x)|^2 dx = 1 ]

• Calculate the integral, which leads to:

  [ A = \left(\frac{2\alpha}{\pi}\right)^{1/4} ]

Step 4: Calculate the Expectation Value of the Energy for the Linear Potential

1. Substitute the Gaussian trial function into the Hamiltonian ( H ) for the linear potential ( V(x) = \alpha |x| ).

2. The expectation value of the energy ( E ) is given by:

   [ E = \int_{-\infty}^{\infty} \psi^*(x) H \psi(x) dx ]

3. Compute the kinetic and potential energy contributions separately.

4. Combine the results to express the total energy as a function of the parameter ( \alpha ).

5. Minimize this energy with respect to ( \alpha ) to find the optimal value.

Step 5: Calculate the Expectation Value of the Energy for the Quartic Potential

1. Repeat the process using the quartic potential ( V(x) = \alpha x^4 ).
2. Again, substitute the Gaussian trial function into the Hamiltonian.
3. Calculate the expectation value of the energy as before, focusing on the quartic potential's contribution.
4. Derive the total energy expression in terms of ( \alpha ) and minimize it to find the optimal ( \alpha ).

Conclusion

By utilizing the variational principle and Gaussian trial functions, you can effectively estimate the ground state energy for different potentials in quantum mechanics. Remember to:

• Define your trial function clearly.
• Normalize it properly.
• Calculate expectation values accurately for both kinetic and potential energies.

With these skills, you can approach more complex quantum systems in your studies. For further exploration, consider applying these methods to other potentials or refining your trial functions for better accuracy.