# 8: Variational Method II (Worksheet)

$$\newcommand{\vecs}{\overset { \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}{\| #1 \|}$$ $$\newcommand{\inner}{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$$$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

Name: ______________________________

Section: _____________________________

Student ID#:__________________________

Work in groups on these problems. You should try to answer the questions without referring to your textbook. If you get stuck, try asking another group for help.

## Constructing the Variational Energy

The variational method is one way of finding approximations to the lowest energy eigenstate or ground state. The method consists of constructing a "trial wavefunction" depending on one or more parameters (e.g., $$|\phi(\alpha, \beta, \gamma, \ldots) \rangle$$ and then evaluating the "trial energy" (variational energy)

$E_\phi (\alpha, \beta, \gamma, \ldots)=\dfrac{\langle\phi (\alpha, \beta, \gamma, \ldots) |\hat H|\phi (\alpha, \beta, \gamma, \ldots) \rangle}{\langle\phi (\alpha, \beta, \gamma, \ldots) |\phi (\alpha, \beta, \gamma, \ldots) \rangle} \label{W1}$

where the Hamiltonian for the system. The wavefunction obtained by fixing the parameters to such values is then an approximation to the true wavefunction.

## Q1

What are the limits on the number of parameters that a trial wavefunction ($$|\phi \rangle$$ can have?

## Q2

Identify which is hardest and why:

• Constructing the Hamiltonian for the system,
• Constructing the trial wavefunction for the system, or
• Evaluating the variational energy.

## Q3

What is the origin of the denominator in Equation \ref{W1}? Does it always have to be in the equation? If not, when can you ignore it?

## Variational Theorem

The variational theorem argues that this trial energy, $$E_\phi$$ associated with the trial wavefunction for the known Hamiltonian is always greater than or equal to the true energy ($$E_\psi$$). Proof is not given.

$E_\phi (\alpha, \beta, \gamma, \ldots) \ge E_\psi \label{VM}$

The variation method approximates the lowest energy eigenvalue, $$E_\psi$$, and eigenfunction, $$\psi$$, for a quantum mechanical system by guessing a function that is well-behaved over the limits of the system and minimizing the energy.

## Q4

Under what condition(s) will this equation be true?

$E_\phi (\alpha, \beta, \gamma, \ldots) = E_\psi \label{VM1}$

## Q5

Explain the power (utility) of Equation \ref{VM}. If this equation were not true, would we be able to approximate the true solutions to the Hamiltonian of the unsolveable system using the variational method?

## Minimizing the Variational Energy

The variational theory argues that when the energy is minimized, then

$E_\phi (\alpha, \beta, \gamma, \ldots) \approx E_{actual}$

and

$| \phi (\alpha, \beta, \gamma, \ldots) \rangle \approx | \psi \rangle$

The better the trial wavefunction resembles the true wavefunction, the more accurate these approximations are.

## Q6

Since one does not know the true eigenstate $$| \psi \rangle$$, how would one conclude that the optimized trial wavefunction is a good approximation to the true wavefunction?

## Continued Application to the Harmonic Oscillator Hamilitonian

From the previous groupwork, we explore the trial wavefunction

$| \phi(x) (\beta) \rangle =\dfrac{1}{1+\beta x^2} \label{trial}$

where $$|\phi \rangle$$ is the wavefunction that we guess and $$\hat H$$ is the Hamiltonian for the system. The variational energy (from solving Equation \ref{W1}) was shown to be

$E_\phi (\beta) =\dfrac{\langle\phi|\hat H|\phi\rangle}{\langle\phi |\phi\rangle}=\dfrac{\hbar^2\beta}{4\mu}+\dfrac{k}{2\beta}$

Some equations you may find useful for the following discussion:

$\hat H_{HO}=-\dfrac{\hbar^2}{2\mu}\dfrac{d^2}{dx^2}+\dfrac{kx^2}{2}$

$E_{n,HO}=h\nu \left(n+\dfrac{1}{2} \right)=\hbar \omega \left(n+\dfrac{1}{2}\right)$

$\underset{\text{lowest energy eigenstate}}{| \psi(x) \rangle}=\left(\dfrac{a}{\pi}\right)^{1/4}e^{-ax^2/2} ﻿﻿$

## Q7

Is the trial wavefunction (Equation \ref{trial}) normalized? And does it matter? Why or why not?

## Q8

What is $$\dfrac{dE_\phi (\beta)}{d\beta}=0$$ and why do you need to evaluate this derivative?

## Q9

What value for $$\beta$$ fulfills the minimized $$E_\phi$$?

## Q10

What is $$E_\phi(\beta)$$ for the value for $$\beta$$ that fulfills the minimization?

## Q11

The variation method approximates the ground state energy for the system. What is the expression for the exact energy of the harmonic oscillator?

## Q12

What is the value for the quantum number for the ground state of the harmonic oscillator?

## Q13

What is the exact energy for the ground state of the harmonic oscillator?

## Q14

Considering only energies, how well does the optimizing trial wavefunction (Equation \ref{trial}) approximate the lowest energy harmonic oscillator eigenstate?

## Q15

How could you improve this approximation?

## Q16

Given the knowledge you have of the true harmonic oscillator wavefunction, how well would a different trial wavefunction, $$|\phi(x) (\beta) \rangle =e^{-\beta x^2}$$ approximate the solution for the lowest energy state of the harmonic oscillator?

This page titled 8: Variational Method II (Worksheet) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Nancy Levinger.