We will show that the components of the angular momentum operator \(\vec{L}\), written in differential operator form in rectangular components, satisfy the commutation relations: \begin{equation} \left[L_x,L_y\right]=+i\hbar L_z \qquad(\text{and cyclic permutations}) \end{equation}
First calculate the components of angular momentum classically: \begin{align} \vec{L}&=\vec{r}\times\vec{p}\\ &=(x\hat{x}+y\hat{y}+z\hat{z})\times(p_x\hat{x}+p_y\hat{y}+p_z\hat{z})\\ &=(yp_z-zp_y)\hat{x}+(zp_x-xp_z)\hat{y}+(xp_y-yp_x)\hat{z} \end{align}
Making the standard quantum substitutions, \begin{align} p_x&\rightarrow -i\hbar\partial_x\\ p_y&\rightarrow -i\hbar\partial_y\\ p_z&\rightarrow -i\hbar\partial_z\\ \end{align} we obtain the following operators for the components of angular momentum: \begin{align} \hat{L}_x&=-i\hbar(y\partial_z-z\partial_y)\\ \hat{L}_y&=-i\hbar(z\partial_x-x\partial_z)\\ \hat{L}_z&=-i\hbar(x\partial_y-y\partial_x)\\ \end{align}
To see the role of the product rule in the commutation relations, it is helpful to give the partial derivatives an arbitrary function \(\psi\) to act on. \begin{align} \left[\hat{L}_x,\hat{L}_y\right]\psi &=\left[-i\hbar(y\partial_z-z\partial_y), -i\hbar(z\partial_x-x\partial_z)\right]\psi\\ &=-\hbar^2\left\{(y\partial_z-z\partial_y)(z\partial_x-x\partial_z) -(z\partial_x-x\partial_z)(y\partial_z-z\partial_y)\right\}\psi \end{align} Now, foil-like-mad. Make sure that all of the partial derivatives act on EVERYTHING to their right. Two of the terms above of the form \begin{align} y\,\partial_z(z\,\partial_x \psi) \end{align} require a product rule: \begin{align} y\,\partial_z(z\,\partial_x \psi) &=y((\partial_z z)(\partial_x\psi)+z(\partial_x\partial_x\psi))\\ &=y\partial_x\psi+yz(\partial_x\partial_x\psi) \end{align}
Continuing the calculation above, we see that all of the second derivative terms will cancel because the order of differentiation doesn't matter, leaving only the first derivative terms from the product rule. \begin{align} \left[\hat{L}_x,\hat{L}_y\right]\psi &=\left[-i\hbar(y\partial_z-z\partial_y), -i\hbar(z\partial_x-x\partial_z)\right]\psi\\ &=-\hbar^2\left\{(y\partial_z-z\partial_y)(z\partial_x-x\partial_z) -(z\partial_x-x\partial_z)(y\partial_z-z\partial_y)\right\}\psi\\ &=-\hbar^2\left\{\left(y\,\partial_z(z\,\partial_x \psi) -y\,\partial_z(x\,\partial_z \psi) -z\,\partial_y(z\,\partial_x \psi) +z\,\partial_y(x\,\partial_z \psi)\right)\right.\\ &\;\;\;\quad\quad\left.-\left(z\,\partial_x(y\,\partial_z \psi) -z\,\partial_x(z\,\partial_y \psi) -x\,\partial_z(y\,\partial_z \psi) +x\,\partial_z(z\,\partial_y \psi)\right) \right\}\\ &=-\hbar^2\left\{\left(\cancel{yz(\partial_z\partial_x\psi)} +y\partial_x\psi -\cancel{yx(\partial_z^2\psi)} -\cancel{z^2(\partial_y\partial_x\psi)} +\cancel{zx(\partial_y\partial_z\psi)}\right)\right.\\ &\;\;\;\quad\quad\left.-\left(\cancel{zy(\partial_x\partial_z\psi)} -\cancel{z^2(\partial_x\partial_y\psi)} -\cancel{xy(\partial_z^2\psi)} +\cancel{xz(\partial_z\partial_y\psi)} +x\partial_y\psi\right) \right\}\\ &=i\hbar\left(-i\hbar(-y\partial_x+x\partial_y)\psi\right)\\ &=i\hbar\hat{L}_z\, \psi \end{align} The other components are cyclic permutations of this calculation.
keyboard Computational Activity
120 min.
finite difference hamiltonian quantum mechanics particle in a box eigenfunctions
Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then usenumpy
to solve for eigenvalues and eigenstates, which they visualize.
assignment Homework
group Small Group Activity
30 min.
face Lecture
120 min.
ideal gas particle in a box grand canonical ensemble chemical potential statistical mechanics
These notes from week 6 of Thermal and Statistical Physics cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.assignment Homework
Consider the following wave functions (over all space - not the infinite square well!):
\(\psi_a(x) = A e^{-x^2/3}\)
\(\psi_b(x) = B \frac{1}{x^2+2} \)
\(\psi_c(x) = C \;\mbox{sech}\left(\frac{x}{5}\right)\) (“sech” is the hyperbolic secant function.)
In each case:
assignment Homework
Given the polar basis kets written as a superposition of Cartesian kets \begin{eqnarray*} \left|{\hat{s}}\right\rangle &=& \cos\phi \left|{\hat{x}}\right\rangle + \sin\phi \left|{\hat{y}}\right\rangle \\ \left|{\hat{\phi}}\right\rangle &=& -\sin\phi \left|{\hat{x}}\right\rangle + \cos\phi \left|{\hat{y}}\right\rangle \end{eqnarray*}
Find the following quantities: \[\left\langle {\hat{x}}\middle|{\hat{s}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{{\hat{s}}}\right\rangle ,\quad \left\langle {\hat{x}}\middle|{\hat{\phi}}\right\rangle ,\quad \left\langle {\hat{y}}\middle|{\hat{\phi}}\right\rangle \]
keyboard Computational Activity
120 min.
inner product wave function quantum mechanics particle in a box
Students compute inner products to expand a wave function in a sinusoidal basis set. This activity introduces the inner product for wave functions, and the idea of approximating a wave function using a finite set of basis functions.group Small Group Activity
30 min.
central forces quantum mechanics eigenstates eigenvalues angular momentum time dependence hermitian operators probability degeneracy quantum measurements
Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.assignment Homework
group Small Group Activity
30 min.