# Worksheet 8: Variational Method II

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*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?