Optimizing quantum ground state energies using the variational method
In my latest study of quantum mechanics, I explore the powerful variational method to approximate the ground state energy of quantum systems where exact solutions are elusive. The technique is used in quantum mechanics for dealing with complex systems, relying on the variational principle which asserts that the energy expectation value calculated with any trial wave function will not undercut the system’s actual ground state energy.
Understanding the variational method
At the core of the variational method is the application of a trial wave function to estimate the ground state energy of a Hamiltonian. Let | \psi_0 \rangle be the exact ground state wave function of the system, with its associated energy eigenvalue E_0. Using the Hermitian property of the Hamiltonian and the wave function’s normalization, we arrive at the relation:
\langle \psi_0 |\mathbf H| \psi_0 \rangle = E_0
However, in cases where | \psi_0 \rangle is not known, I employ a trial wave function | \phi \rangle, a linear combination of a set of eigenfunctions:
| \phi \rangle = \sum_n c_n | \psi_n \rangle
where E_n are the eigenvalues associated with these eigenfunctions. The energy expectation value then becomes:
\langle \phi | \mathbf H | \phi \rangle = \sum_n |c_n|^2 E_n
indicating that any approximation will inherently be greater than or equal to E_0.
Practical application: infinite potential well
One practical application of this method is analyzing an electron in an infinite potential well under an external electric field. The Hamiltonian for such a system is given by:
\mathbf H = -\frac{1}{\pi^2}\frac{\mathrm d^2}{\mathrm d\xi^2} + f\left(\xi-\frac{1}{2}\right)
I choose a trial function, \phi_\text{trial}(\xi), that combines the first two states of the unperturbed well, ensuring normalization:
\phi_\text{trial}(\xi) = \frac{\sqrt 2}{\sqrt{1+ \alpha^2}} \left[\sin\left(\pi\xi\right) + \alpha\sin\left(2\pi\xi\right)\right]
This setup allows me to calculate the energy expectation value:
\langle \phi_\text{trial} | \mathbf H | \phi_\text{trial} \rangle = \frac{1}{1+ \alpha^2} \left( \varepsilon_1 + 4\alpha^2\varepsilon_1 - \frac{32f\alpha}{9\pi^2} \right)
Optimizing the trial function
To find the minimum energy configuration, the derivative of the expectation value with respect to the variational parameter \alpha is computed, leading to a condition that determines the best \alpha value:
\frac{\mathrm{d} \langle \phi_\text{trial} | \mathbf H | \phi_\text{trial} \rangle}{\mathrm{d}\alpha} = 0
Solving this equation provides the optimal \alpha, hence approximating the lowest energy state of the electron in the well under the influence of an electric field. The results show excellent agreement with known analytical and numerical solutions, demonstrating the variational method’s efficacy in this quantum system.
Conclusion
The variational method proves invaluable for estimating the ground state energies of quantum systems, providing a reliable approach when exact methods falter. My analysis here underscores its potential and adaptability to different quantum scenarios. For more insights into this topic, you can find the details here.