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May 9th 2008, 09:49 AM
DiscreteW

Qm

Hey topsquark,

Huge Quantum exam tomorrow! Few practice probs I didn't understand:

1.) Suppose that an atom in a solid is described by $H = H_0 + \beta x^2$ , where $H$ is the hamilotonian, and $H_0 ~\mbox{is the simple harmonic oscillators Hamiltonian}$ . Also $\beta x^2$ is needed for the vibrations of an atom (in an object). $\beta$ is a real constant. Determine the $n$ -th energy eigenvalue $E_n$ of $H$ , assuming $H_0$ 's $n$ -th eigenvalue $U_n>$ is also the $n$ -th[/tex] energy eigenstate of $H = H_0 + \beta x^2$ .

2a.) Using quantum mechanics, describe how you'd solve the system of a hydrogen atom. What are the physical results?

b.) Use separation of variables in $(x,y)$ coordinates in order to solve the time-indep. schrodinger equation for a particle (free) that's restricted to a 2-D square box. The PE (potential) of this system is:

$V(x,y) = 0,~ \mbox{when both x and y are between 0 and a}$ .
$........... = \infty,~ \mbox{elsewhere (thus the particle is not able to leave the box)}$

c.) Determine all energy eigenfuntions and their energy eigenvalues. Then, determine what the degeneracy of the energy state $E_2$ . (Note, $E_1$ is the lowest level).
May 9th 2008, 11:32 AM
DiscreteW

The hint for #1 was to express $x^2$ in terms of $a$ and $a\dagger$ .
May 9th 2008, 11:50 AM
topsquark

I can't get to these right now. (If I even tried I'd probably fall asleep at the keyboard!) If I can't get to them tonight I will try to get to them before the weekend is over.

-Dan
May 9th 2008, 03:39 PM
DiscreteW

I worked on 2b somewhat..

$\hat{H} = \frac{-\hbar}{2m}\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) + V(x,y)$

Separation of variables: let $\psi(x,y) = X(x)Y(y)$

$\hat{H}\psi(x,y) = E\psi(x,y)$

$\hat{H}X(x)Y(y) = \frac{-\hbar^2}{2m}\left(\frac{\partial ^2}{\partial x^2}X(x)Y(y) + \frac{\partial^2}{\partial y^2}X(x)Y(y)\right)$
$+ V(x,y)X(x)Y(y) = EX(x)Y(x)$

$\frac{-\hbar^2}{2m}\left(Y(y)\frac{d^2}{dx^2}X(x) + X(x)\frac{d^2}{dy^2}Y(y)\right) + (V(x,y) - E)X(x)Y(y) = 0$

$Y(y)\frac{d^2}{dx^2}X(x) + X(x)\frac{d^2}{dy^2}Y(y) + \frac{2m}{\hbar^2}\left(E-V(x,y)\right)X(x)Y(y) = 0$

Almost..
May 9th 2008, 04:32 PM
mr fantastic

Quote:

Originally Posted by DiscreteW

Hey topsquark,

Huge Quantum exam tomorrow! Few practice probs I didn't understand:

1.) Suppose that an atom in a solid is described by $H = H_0 + \beta x^2$ , where $H$ is the hamilotonian, and $H_0 ~\mbox{is the simple harmonic oscillators Hamiltonian}$ . Also $\beta x^2$ is needed for the vibrations of an atom (in an object). $\beta$ is a real constant. Determine the $n$ -th energy eigenvalue $E_n$ of $H$ , assuming $H_0$ 's $n$ -th eigenvalue $U_n>$ is also the $n$ -th[/tex] energy eigenstate of $H = H_0 + \beta x^2$ .

2a.) Using quantum mechanics, describe how you'd solve the system of a hydrogen atom. What are the physical results? Mr F says: This is most certainly found in any quantum mechanics textbook .....

b.) Use separation of variables in $(x,y)$ coordinates in order to solve the time-indep. schrodinger equation for a particle (free) that's restricted to a 2-D square box. The PE (potential) of this system is:

$V(x,y) = 0,~ \mbox{when both x and y are between 0 and a}$ .
$........... = \infty,~ \mbox{elsewhere (thus the particle is not able to leave the box)}$

c.) Determine all energy eigenfuntions and their energy eigenvalues. Then, determine what the degeneracy of the energy state $E_2$ . (Note, $E_1$ is the lowest level).

I don't want to rain on anyone's parade, but the solutions to all these questions can be found as worked examples in most quantum mechanics textbooks.

For 1.), since $H_0 = \frac{p_x^2}{2m} + \frac{m \omega^2}{2} \, x^2$ , you could write $H = \frac{p_x^2}{2m} + \left( \frac{m \omega^2}{2} + \beta \right) x^2 = \frac{p_x^2}{2m} + \frac{m (\omega ')^2}{2} \, x^2$ and treat it as a one-dimensional harmonic oscillator (the detailed solution to which is given in all quantum mechanics textbooks - spanning several pages).

Read this: Quantum harmonic oscillator - Wikipedia, the free encyclopedia

Or this: Quantum mechanics/harmonic oscillator/operator method - Physics

-------------------------------------------------------------------------------------------------------

For 2b.), you have a particle confined to a two-dimensional infinite square well. Since $V(x, y) = 0$ for $0 < x < a$ and $0 < y < a$ , the Schroedinger equation is:

$\frac{-\hbar}{2m}\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) \psi(x, y) = E \psi(x, y)$ .

Read this: Square Wells p.7

Or this: Particle in a box - Wikipedia, the free encyclopedia
May 9th 2008, 06:04 PM
DiscreteW

For #1,

$\hat{H} = \hbar E_n\left(\omega + \frac{2\beta}{m\omega}\right)$

If this is the sol’n then I understand it, if not then I have no idea.
May 9th 2008, 07:04 PM
mr fantastic

Quote:

Originally Posted by DiscreteW

For #1,

$\hat{H} = \hbar E_n\left(\omega + \frac{2\beta}{m\omega}\right)$

If this is the sol’n then I understand it, if not then I have no idea.

The eigen values of H are:

$E_n = \hbar \omega' (n + 1/2)$ where $\omega ' = \sqrt{\omega^2 + 2\beta /m}$

I would have thought.

I'll leave it to topsquark to comment further when he gets the chance.
May 10th 2008, 02:45 AM
topsquark

Quote:

Originally Posted by DiscreteW

1.) Suppose that an atom in a solid is described by $H = H_0 + \beta x^2$ , where $H$ is the hamilotonian, and $H_0 ~\mbox{is the simple harmonic oscillators Hamiltonian}$ . Also $\beta x^2$ is needed for the vibrations of an atom (in an object). $\beta$ is a real constant. Determine the $n$ -th energy eigenvalue $E_n$ of $H$ , assuming $H_0$ 's $n$ -th eigenvalue $U_n>$ is also the $n$ -th[/tex] energy eigenstate of $H = H_0 + \beta x^2$ .

Quote:

Originally Posted by mr fantastic

The eigen values of H are:

$E_n = \hbar \omega' (n + 1/2)$ where $\omega ' = \sqrt{\omega^2 + 2\beta /m}$

Mr. Fanastic is correct here.

Is this actually supposed to be a perturbation problem? (ie. $\beta$ is a small number and we want the approximate solution.) That's what the setup looks like, anyway.

-Dan
May 10th 2008, 02:49 AM
topsquark

Quote:

Originally Posted by DiscreteW

2a.) Using quantum mechanics, describe how you'd solve the system of a hydrogen atom. What are the physical results?

This is far too complicated a task to write down in a (single) post. The plan of attack:
The wavefunction in spherical-polar coordinates can be separated into a radial (r) and angular ( $\theta, \phi$ ) parts.

The radial wavefunction solution involves LaGuerre polynomials and the angular part solution deals with spherical harmonics. (The angular part of the wavefunction is also separable: the $/phi$ solution is merely harmonic, and the $\theta$ solution deals with associated Legendre polynomials.)

-Dan
May 10th 2008, 02:53 AM
topsquark

Quote:

Originally Posted by DiscreteW

I worked on 2b somewhat..

$\hat{H} = \frac{-\hbar}{2m}\left(\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2}\right) + V(x,y)$

Separation of variables: let $\psi(x,y) = X(x)Y(y)$

$\hat{H}\psi(x,y) = E\psi(x,y)$

$\hat{H}X(x)Y(y) = \frac{-\hbar^2}{2m}\left(\frac{\partial ^2}{\partial x^2}X(x)Y(y) + \frac{\partial^2}{\partial y^2}X(x)Y(y)\right)$
$+ V(x,y)X(x)Y(y) = EX(x)Y(x)$

$\frac{-\hbar^2}{2m}\left(Y(y)\frac{d^2}{dx^2}X(x) + X(x)\frac{d^2}{dy^2}Y(y)\right) + (V(x,y) - E)X(x)Y(y) = 0$

$Y(y)\frac{d^2}{dx^2}X(x) + X(x)\frac{d^2}{dy^2}Y(y) + \frac{2m}{\hbar^2}\left(E-V(x,y)\right)X(x)Y(y) = 0$

Almost..

Divide the whole thing by $X(x)Y(y)$ .
$\frac{1}{X}\frac{d^2}{dx^2}X(x) + \frac{1}{Y}\frac{d^2}{dy^2}Y(y) = -\frac{2m}{\hbar^2}\left(E-V(x,y)\right)$

In the box, the potential is 0:
$\frac{1}{X}\frac{d^2}{dx^2}X(x) + \frac{1}{Y}\frac{d^2}{dy^2}Y(y) = -\frac{2m}{\hbar^2}E = \text{constant}$

Thus the X and Y parts of the equation are merely constants themselves (otherwise they would depend on both x and y.) Call
$\frac{1}{X}\frac{d^2}{dx^2}X(x) = -k_x^2$
and
$\frac{1}{Y}\frac{d^2}{dy^2}Y(y) = -k_y^2$
then (of course)
$k_x^2 + k_y^2 = \frac{2m}{\hbar^2}E$

and go from there.

-Dan
May 10th 2008, 03:02 AM
topsquark

Quote:

Originally Posted by DiscreteW

c.) Determine all energy eigenfuntions and their energy eigenvalues. Then, determine what the degeneracy of the energy state $E_2$ . (Note, $E_1$ is the lowest level).

You will find that
$E_{nm} = \frac{\pi ^2 \hbar ^2}{2ma^2}(n^2 + m^2)$
(where n and m denote the constants for the x and y parts of the solution.)

The problem is denoting the energy levels as E1, E2, ... in order of increasing energy. So
$E_1 = E_{10} = E_{01} = \frac{\pi ^2 \hbar ^2}{2ma^2}$

$E_2 = E_{11} = \frac{2\pi ^2 \hbar ^2}{2ma^2}$

$E_3 = E_{20} = E_{02} = \frac{4\pi ^2 \hbar ^2}{2ma^2}$

etc.

Carefully note some "strange" features of the energy spectrum such as
$E_{33} > E_{40}$
so be careful of simply using the "obvious" pattern to decide which energy level is higher than another.

-Dan