Choosing the trial wavefunction (variational method)

[rwooduk]

Homework Statement

I'm going to list two questions as they offer the same problem with more choices, hopefully it will help realize the method (?) used

(A)
An electron, confined in the two dimensional region 0<x<L and 0<y<L with infinite potential walls, is subject to the potential:

$$V(x,y)=V_{1}x + V_{2}y^{2}$$

Which of the following trial wave functions is better suited for approximating the wave function of the electron:

$$\psi_{T} (x,y) = cos(ax+by)$$
$$\psi_{T} (x,y) = exp -(ax^{2} + by^{4})$$
$$\psi_{T} (x,y) = xy(a-x)(b-y)$$

(B)
Consider the infinite potential well V(x) = ∞ at x = ±L/2 and for -L/2 < x < L/2 it is:

$$V(x) = \frac{1}{2} k(x-x_{0})^2$$

Which trial wavefunction should we consider:

$$\psi_{T} (x) = x(x-L)exp -\gamma(x-x^{0})^2$$
$$\psi_{T} (x) = (x- \frac{L}{2} )(x+ \frac{L}{2} )exp -\gamma(x-x_{0})^2$$
$$\psi_{T} (x) = \alpha(x- \frac{L}{2} )(x+ \frac{L}{2} ) + \beta exp -\gamma (x-x_{0})^2$$
$$\psi_{T} (x) = (x- \frac{L}{2} )^{\alpha}(x+ \frac{L}{2} )^{\beta} + \beta exp -\gamma (x-x_{0})^2$$

More examples from other questions, where the trial wavefunction was already "chosen"

$$if... V(r) = A exp (- \frac{r}{a} ) ----> \psi_{T} = c exp (- \frac{\alpha r}{2a} )$$
$$if... V(r)= \frac{V_{0} x}{L} ----> \psi_{T} = x(x-L)$$
$$if... V(r)=0 ----> \psi_{T} = sin(ax+b)$$

Homework Equations

Intuition? Boundary conditions? Should it be of the same form as the potential?

The Attempt at a Solution

Sometimes it seems the trial wavefunction is picked at random out of a hat, I really don't see how it is chosen. Is there a general method? I asked the lecturer and he said to look at the boundary conditions for the problem, but how would that help me decide?

I really think this question could help a lot of people as I've searched on the internet and most of the time it's simply defined.

Any ideas would be welcome.

 

[TSny]

Can you state the boundary conditions on the wave function for (A) and (B)?

 

[rwooduk]

Can you state the boundary conditions on the wave function for (A) and (B)?

for Question (A) and (B) its an infinite well so I'd assume the boundary conditions to be ψ(0)=ψ(L)=0 that is zero at the boundaries. but can't see how that would help determine the trial wave function.

many thanks for the reply

 

[TSny]

Right. The wavefunction must vanish at each point of the boundary. Note that in part (A) the wavefunction is a function of both x and y: $\psi(x,y)$.

Decide what values of x and y will correspond to being on the boundary of the potential well. Then check each proposed wavefunction to see if the wavefunction vanishes for those values of x and y.

Something looks a little odd in part (A). The statement of the problems specifies 0 < x < L and 0 < y < L. However, the wavefunctions do not contain the constant L. Instead they contain the two constants a and b. I wonder if the statement of the problem should have specified the region as 0 < x < a and 0 < y < b.

 

[rwooduk]

Right. The wavefunction must vanish at each point of the boundary. Note that in part (A) the wavefunction is a function of both x and y: $\psi(x,y)$.

Decide what values of x and y will correspond to being on the boundary of the potential well. Then check each proposed wavefunction to see if the wavefunction vanishes for those values of x and y.

Thank you I will try this.

Something looks a little odd in part (A). The statement of the problems specifies 0 < x < L and 0 < y < L. However, the wavefunctions do not contain the constant L. Instead they contain the two constants a and b. I wonder if the statement of the problem should have specified the region as 0 < x < a and 0 < y < b.

Ive double checked the question and it's correct, later in the question it does say a and b are free parameters. Here's a link to the paper for your own interest:

https://www.leeds.ac.uk/students/office/exampapers/webserv3_exampapers/phas/13PHYS338101s1.pdf

It's question B2 (c)

Thanks for your help!

 

[TSny]

My understanding of the variation method is to choose a trial wavefunction that satisfies the boundary conditions and also has at least one free parameter to vary. In part (A), it appears to me that only one of the proposed functions can satisfy the boundary conditions once a and b are chosen appropriately. But then there will be no free parameter for the variation procedure. Hope I'm not overlooking something. The (B) part looks OK. I think there is one and only one valid choice here.

 

[rwooduk]

Ok, I'm going to try complete problem (A) (including deriving E_T) for anyone else dealing with something similar. I'm just stuck on how to combine the answers at the end. Here's what I've got:

Using the given boundary conditions:

$$\Psi _{T}(0,y)=\Psi _{T}(L,y)=0$$
$$\Psi _{T}(x,0)=\Psi _{T}(x,L)=0$$

For:

$$\psi_{T} (x,y) = cos(ax+by)$$

Applying the boundary conditions gives a=0 and b=0 so this is not appropriate.

For:

$$\psi_{T} (x,y) = exp -(ax^{2} + by^{4})$$

Applying the boundary conditions gives a=0 and b=0 so this is not appropriate.

For:

$$\psi_{T} (x,y) = xy(a-x)(b-y)$$

Applying the boundary conditions when x=0 Ψ=0 and when y=0 Ψ=0, therefore this is the trial wave function we must use.

Now we define the equation for determining the energy

$$E_{T}= \frac{\left \langle \Psi _{T}(x) |H |\Psi _{T}(x) \right \rangle}{\left \langle \Psi _{T}(x) |\Psi _{T}(x) \right \rangle}= \frac{\left \langle \Psi _{T}(x) |T |\Psi _{T}(x) \right \rangle + \left \langle \Psi _{T}(x) |V |\Psi _{T}(x) \right \rangle}{\left \langle \Psi _{T}(x) |\Psi _{T}(x) \right \rangle}$$

The question states as a hint to use separation of variables so we can write

$$\psi_{T} (x,y) = xy(a-x)(b-y)$$

as

$$X_{T}(x)= x(a-x)$$
$$Y_{T}(y)= y(b-y)$$

Since the equations are similar we can calculate E_T for X_T(x) and then by comparison determine E_T for Y_T(y), NB the potential term for y is different! These are my results:

$$\left \langle X_{T} | X_{T} \right \rangle =\frac{L^{5}}{5}-\frac{a^{2}L^{3}}{3}$$
$$\left \langle Y_{T} | Y_{T} \right \rangle =\frac{L^{5}}{5}-\frac{b^{2}L^{3}}{3}$$
$$\left \langle X_{T} |T| X_{T} \right \rangle =\frac{\hbar}{m}(\frac{aL^{2}}{2}-\frac{L^{3}}{3})$$
$$\left \langle Y_{T} |T| Y_{T} \right \rangle =\frac{\hbar}{m}(\frac{bL^{2}}{2}-\frac{L^{3}}{3})$$
$$\left \langle X_{T} |V_{x}| X_{T} \right \rangle = (\frac{L^{6}}{6}-\frac{a^{2}L^{4}}{4})V_{1}$$
$$\left \langle Y_{T} |V_{y}| Y_{T} \right \rangle = (\frac{L^{7}}{7}-\frac{b^{2}L^{5}}{5}-\frac{2bL^{6}}{6})V_{2}$$

Question how do I insert these terms in the original equation:

$$E_{T}= \frac{\left \langle \Psi _{T}(x) |H |\Psi _{T}(x) \right \rangle}{\left \langle \Psi _{T}(x) |\Psi _{T}(x) \right \rangle}= \frac{\left \langle \Psi _{T}(x) |T |\Psi _{T}(x) \right \rangle + \left \langle \Psi _{T}(x) |V |\Psi _{T}(x) \right \rangle}{\left \langle \Psi _{T}(x) |\Psi _{T}(x) \right \rangle}$$