Probability analogy

• For independent random variables: \(P(X=x,Y=y)=P_x(X=x)\,P_y(Y=y)\).

• E.g. coin + die: \(P(H,6)=\tfrac12\cdot\tfrac16=\tfrac1{12}\).

• Likewise \(\psi(x,y)=X(x)Y(y)\), with \(\int|X|^2dx=\int|Y|^2dy=1\) so that \(\iint|\psi|^2dx\,dy=1\).

• Number of generalized coordinates = degrees of freedom = number of quantum numbers.

Particle in a Two-Dimensional Infinite Box

• \(V(x,y)=0\) for \(0<x<a\), \(0<y<b\); \(\infty\) otherwise.   \(H=\dfrac{P_x^2}{2m}+\dfrac{P_y^2}{2m}\).

• Substitute \(\psi=X(x)Y(y)\) and divide by \(XY\): \[-\frac{\hbar^2}{2m}\frac{1}{X}\frac{d^2X}{dx^2}-\frac{\hbar^2}{2m}\frac1Y\frac{d^2Y}{dy^2}=E\;\Rightarrow\;E=E_x+E_y.\]

• Each part is a 1D box: \(X''+k_x^2X=0\), \(X(0)=X(a)=0\) \(\Rightarrow k_x=\dfrac{n_x\pi}{a}\), \(X=\sqrt{\dfrac2a}\sin\dfrac{n_x\pi x}{a}\) (similarly for \(Y\)).

• Result (\(n_x,n_y=1,2,3,\ldots\)): \[\psi_{n_xn_y}=\sqrt{\frac2a}\sin\frac{n_x\pi x}{a}\sqrt{\frac2b}\sin\frac{n_y\pi y}{b},\qquad E_{n_xn_y}=\frac{\pi^2\hbar^2}{2m}\Big(\frac{n_x^2}{a^2}+\frac{n_y^2}{b^2}\Big).\]

Free Particle in Two and Three Dimensions

• 2D: \(\psi=A\big(e^{ik_xx}+e^{-ik_xx}\big)\big(e^{ik_yy}+e^{-ik_yy}\big)\),   \(E=\dfrac{\hbar^2}{2m}(k_x^2+k_y^2)=\dfrac{\hbar^2k^2}{2m}\).

• 3D: \(E=\dfrac{\hbar^2}{2m}(k_x^2+k_y^2+k_z^2)\).

• Any \(\vec k\) of fixed magnitude gives the same \(E\) \(\Rightarrow\) infinitely degenerate.

Two-Dimensional Square Box

• \(a=b\):   \(E=\dfrac{\pi^2\hbar^2}{2ma^2}(n_x^2+n_y^2)\equiv(n_x^2+n_y^2)E_0\).

Three-Dimensional Box

• \(V=0\) inside \(L_x\times L_y\times L_z\), \(\infty\) outside: \[\psi=\sqrt{\frac{8}{L_xL_yL_z}}\sin\frac{n_x\pi x}{L_x}\sin\frac{n_y\pi y}{L_y}\sin\frac{n_z\pi z}{L_z},\qquad E=\frac{\pi^2\hbar^2}{2m}\Big(\frac{n_x^2}{L_x^2}+\frac{n_y^2}{L_y^2}+\frac{n_z^2}{L_z^2}\Big).\]

• Cubic box (\(L\)): \(E=(n_x^2+n_y^2+n_z^2)E_0\), \(E_0=\dfrac{\pi^2\hbar^2}{2mL^2}\).