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

    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 $\displaystyle (x,y)$ 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:

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

    c.) Determine all energy eigenfuntions and their energy eigenvalues. Then, determine what the degeneracy of the energy state $\displaystyle E_2$. (Note, $\displaystyle E_1$ is the lowest level).
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  2. #2
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    The hint for #1 was to express $\displaystyle x^2$ in terms of $\displaystyle a$ and $\displaystyle a\dagger$.
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    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
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  4. #4
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    I worked on 2b somewhat..

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


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


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


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


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


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

    Almost..
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  5. #5
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    Quote Originally Posted by DiscreteW View Post
    Hey topsquark,

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

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

    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 $\displaystyle (x,y)$ 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:

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

    c.) Determine all energy eigenfuntions and their energy eigenvalues. Then, determine what the degeneracy of the energy state $\displaystyle E_2$. (Note, $\displaystyle E_1$ 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 $\displaystyle H_0 = \frac{p_x^2}{2m} + \frac{m \omega^2}{2} \, x^2$, you could write $\displaystyle H = \frac{p_x^2}{2m} + \left( \frac{m \omega^2}{2} + \beta \right) x^2 = \frac{p_x^2}{2m} + \frac{m (\omega ')^2}{2} \, x^2$ 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 $\displaystyle V(x, y) = 0$ for $\displaystyle 0 < x < a$ and $\displaystyle 0 < y < a$, the Schroedinger equation is:

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

    Read this: Square Wells p.7

    Or this: Particle in a box - Wikipedia, the free encyclopedia
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    For #1,

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

    If this is the solín then I understand it, if not then I have no idea.
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  7. #7
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    Quote Originally Posted by DiscreteW View Post
    For #1,

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

    If this is the sol’n then I understand it, if not then I have no idea.
    The eigen values of H are:

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

    I would have thought.

    I'll leave it to topsquark to comment further when he gets the chance.
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    Quote Originally Posted by DiscreteW View Post
    1.) Suppose that an atom in a solid is described by $\displaystyle H = H_0 + \beta x^2$, where $\displaystyle H$ is the hamilotonian, and $\displaystyle H_0 ~\mbox{is the simple harmonic oscillators Hamiltonian}$. Also $\displaystyle \beta x^2$ is needed for the vibrations of an atom (in an object). $\displaystyle \beta$ is a real constant. Determine the $\displaystyle n$-th energy eigenvalue $\displaystyle E_n$ of $\displaystyle H$, assuming $\displaystyle H_0$ 's $\displaystyle n$-th eigenvalue $\displaystyle U_n>$ is also the $\displaystyle n$-th[/tex] energy eigenstate of $\displaystyle H = H_0 + \beta x^2$.
    Quote Originally Posted by mr fantastic View Post
    The eigen values of H are:

    $\displaystyle E_n = \hbar \omega' (n + 1/2)$ where $\displaystyle \omega ' = \sqrt{\omega^2 + 2\beta /m}$
    Mr. Fanastic is correct here.

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

    -Dan
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  9. #9
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by DiscreteW View Post
    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 ($\displaystyle \theta, \phi$) 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 $\displaystyle /phi$ solution is merely harmonic, and the $\displaystyle \theta$ solution deals with associated Legendre polynomials.)

    -Dan
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  10. #10
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    Quote Originally Posted by DiscreteW View Post
    I worked on 2b somewhat..

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


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


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


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


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


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

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

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

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

    and go from there.

    -Dan
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  11. #11
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    Quote Originally Posted by DiscreteW View Post
    c.) Determine all energy eigenfuntions and their energy eigenvalues. Then, determine what the degeneracy of the energy state $\displaystyle E_2$. (Note, $\displaystyle E_1$ is the lowest level).
    You will find that
    $\displaystyle E_{nm} = \frac{\pi ^2 \hbar ^2}{2ma^2}(n^2 + m^2)$
    (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
    $\displaystyle E_1 = E_{10} = E_{01} = \frac{\pi ^2 \hbar ^2}{2ma^2}$

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

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

    etc.

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

    -Dan
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