Isotopic Harmonic Oscillator

Isotopic Harmonic Oscillator:

For isotopic harmonic oscillator

$$\omega= \omega_x= \omega_y=\omega_z$$

The potential energy is given by,

$$v(r)=\frac12 k\cdot r^2$$

$$V(r)=\frac12 \mu\omega^2(x^2+y^2+z^2))\dotsm(1)$$

Where, \(\omega=\sqrt{\frac{k_x}{\mu}}\)

The Schrodinger equation is,

$$\hat H\psi= E\psi\dotsm(2)$$

$$or,\; [\frac{-\hbar^2}{2\mu}\nabla^2+v(\vec r)]\psi= E\psi\dotsm(3)$$

In the cartesian co-ordinate, the Hamiltonian of the system can be written as,

$$H=H_x+H_y+H_z$$

and \(\psi(x,y,z)= l u_x(x), u_y(y), u_z(z)\)

Since the variable can be saperated so,

$$H_x=\biggl[\frac{-\hbar^2}{2\mu}\frac{d^2}{dx^2}+V(x)\biggr]u_x(x)= E_x U_x(x)$$

$$H_y= \biggl[\frac{-\hbar^2}{2\mu}\frac{d^2}{dy^2}+ V(y)\biggr]u_y(y)= E_y u_y(y)$$

$$H_z=\biggl[\frac{-\hbar^2}{2\mu}\frac{d^2}{dz^2}+v(z)\biggr]u_z(z)= E_z u_z(z)$$

and

$$E= \biggl[(n_x+n_y+n_z)+\frac32\biggr]\hbar\omega$$

now, \(\psi_{nx,ny,nz}= U_{nx}- U_{ny}U_{nz}\)

Where,

$$U_{nx}= \biggl(\frac{\alpha_s}{2^n n!\sqrt{\pi}}\biggr)^\frac12 e^{-\alpha_x \frac{x^2}{2}}H_{nx}(\alpha_s, x)$$

Also,,

$$U_{ny}= \biggl(\frac{\alpha_x}{2^n n!\sqrt{\pi}}\biggr)\frac12 e^{-\alpha_x \frac{y^2}{2}} H_{ny}(\alpha_x\cdot z)$$

$$U_{nz}= \biggl(\frac{\alpha_x}{2^n n!\sqrt{\pi}}\biggr)\frac12 e^{-\alpha_s\frac{z^2}{2}}H_{nz}(\alpha_s\cdot z)$$

Where, \(\alpha_s= \sqrt{\frac{\mu \omega}{\hbar}}\)

So, non degenerate quantum zero point energy is \(\frac32\hbar \omega\)

So the degeneracy depends upon \((n_x, n_y n_z)\)

And \(g= \frac12(n+1)(n+2)\)

For n=1, g=3

$$(n_x,n_y,n_z)= (1,0,0)\; (0,1,0)\; (0,0,1)$$

Now, consider a pair of creation and annihilation operator is

$$a_j= \sqrt{\frac{\mu\omega}{2\hbar}}x_j+\frac{i}{\sqrt{2\pi\omega\hbar}}P_j$$

Where, j= x,y,z

And \(x_x=x\;\; x_y= z\;\; x_y=z\;\; x_z= z\)

Now, $$[a_j^+ , a_j]=1\; and \; H=\hbar \omega \sum_{j=1}^3[a_j^+ a_j+\frac32]$$

Now,

$$a_2|\psi_{nx,ny,xz}>= a_x|u_{nx}>|U_{ny}>|U_{n2}>$$

$$=\sqrt{n_x}|U_{nx-1}>|U_{ny}>|U_{nz}>$$

$$=\sqrt{n_x} |\psi_{nx-1, ny,nz}>$$

Also, \(a_x^+|\psi_{nx,ny,nz}>= \sqrt{n_{x+1}}|psi_{nx+1,ny,nz}>\)

So the wave function for isotopic harmonic oscillator in terms, creation and annihilation operator can be written as,

$$|U_{nx}>=\frac{1}{\sqrt{n_x!}}(a_x^+)^{n_z} a_x|u_0>$$

$$|U_{ny}>=\frac{1}{\sqrt{n_y!}}(a_y^+)^n_y a_y|u_0>$$

$$|U_{nz}>=\frac{1}{\sqrt{n_z!}}(a_z^+)n_z a_z|u_0>$$

Now the total wave function can be written as,

$$\psi_{nx,ny,nz}=\frac{1}{\sqrt{n_x!n_y!n_z!}} (a_x^+)^n_x (a_y^+)^n_y (a_z^+)n_z|\psi_{100}>$$

Reference:

1. Mathews, P.M and K Venkatesan.A Text Book of Quantum Mechanics.New Delhi: Tata McGraw Hill Publishing Co. Ltd, 1997.
2. Merzbacher, E.Quantum Mechanics .New York: John Wiley, 1969.
3. Prakash, S and S Salauja.Quantum Mechanics.Kedar Nath Ram Nath Publishing Co, 2002.
4. Singh, S.P, M.K and K Singh.Quantum Mechanics.Chand & Company Ltd., 2002.