1. Introduction
Dynamics play fundamental roles in quantum physics. When an external field is applied to a quantum system, the wave function of the system evolves in time. The properties of the system may then be modified. The field can then be served as a probe to investigate the properties of the system. The coupling of magnetic field and charged particles leads to many interesting phenomena, such as Landau levels, the Aharonov-Bohm effect [1], the integral and fractional quantum Hall effect [2, 3], the Messiner effect [4], etc.
Angular momentum is one of the elementary properties of particles. Recently, much work has been done on angular momentum-related quantum phenomena. It has been found that a magnetic field can modify the angular momentum of an electron [5, 6]. The magnetic field can affect the chemical reaction when the molecules have angular momentum [7]. Despite these processes, a detailed wave function of a particle with angular momentum evolving in a magnetic field is still lacking. In this paper, we have obtained the evolution of eigenfunctions of various angular momenta, and some interesting features have been found. The group velocity of the particle is set to zero. It will experience no Lorentz force in a magnetic field according to classical theory [8]. But in reality, due to the wave quality, there also exists dynamics. Furthermore, for a charged particle with orbital angular momentum, it has the corresponding magnetic momentum [9]. The coupling of the latter with the magnetic field also leads to the dynamics of the particle.
Our main findings are that the uniform magnetic field plays effective roles in two aspects: one is that it serves as a two-dimensional harmonic potential, which makes the wave function oscillate with a time period related to the strength of the potential; the other is that it serves to rotate the wave function with respect to an axis along the direction of the magnetic field, with a time period inverse proportional to the strength of the magnetic field. We also extend the results phenomenanically to the case with an external magnetic field that varies harmonically in time. We argue that the probability density of the eigenfunctions of ${\hat{L}}_{x}$ oscillates between clockwise and anticlockwise with the magnitude of the angular velocity varying harmonically in time.
2. Evolution of eigenfunctions of orbital angular momentum
The evolution of the wave function of a charged particle in a magnetic field is governed by the Hamiltonian1 ) has
$\begin{eqnarray}{\hat{H}}_{0}=\displaystyle \frac{{\left(\hat{{\boldsymbol{p}}}+\tfrac{q}{c}{\boldsymbol{A}}\right)}^{2}}{2M}.\end{eqnarray}$
Here, q is the charge of the particle. c is the velocity of light propagating in a vacuum. M is the mass of the particle. A is the magnetic vector of the magnetic field. The magnetic field is set to be along the z direction with strength B, that is, B = (0, 0, B). In the axis-symmetric gauge with ${\boldsymbol{A}}=\tfrac{1}{2}{\boldsymbol{B}}\times {\boldsymbol{x}}$, the Hamiltonian in equation ( $\begin{eqnarray}\begin{array}{l}{\hat{H}}_{0}=\displaystyle \frac{{\hat{p}}_{x}^{2}+{\hat{p}}_{y}^{2}+{\hat{p}}_{z}^{2}}{2M}\\ \quad +\,\displaystyle \frac{1}{8}M{\omega }_{L}^{2}({x}^{2}+{y}^{2})+\displaystyle \frac{1}{2}{\omega }_{L}{\hat{L}}_{z},\end{array}\end{eqnarray}$
where ${\omega }_{L}=\tfrac{{qB}}{{Mc}}$, and ${\hat{L}}_{z}={\left(\hat{{\boldsymbol{x}}}\times \hat{{\boldsymbol{p}}}\right)}_{z}=\hat{x}{\hat{p}}_{y}-\hat{y}{\hat{p}}_{x}$.The roles played by the magnetic field on the charged particle are effectively in two aspects: a two-dimensional harmonic potential, which confines the wave packet of the particle in the plane perpendicular to the direction of the magnetic field; and an orbital angular momentum term, which rotate wave function around an axis along the direction of the magnetic field.
We also apply a harmonic potential V(x) on the particle with
$\begin{eqnarray}V({\boldsymbol{x}})=\displaystyle \frac{1}{2}M{{\rm{\Omega }}}_{\perp }^{2}({x}^{2}+{y}^{2})+\displaystyle \frac{1}{2}M{{\rm{\Omega }}}_{z}^{2}{z}^{2}.\end{eqnarray}$
For simplicity, we choose the case with ${{\rm{\Omega }}}_{z}=\sqrt{{{\rm{\Omega }}}_{\perp }^{2}+{\omega }_{L}^{2}/4}\,={\rm{\Omega }}/2$. Then the total Hamiltonian is $\begin{eqnarray}\begin{array}{l}\hat{H}=\displaystyle \frac{1}{2M}({\hat{p}}_{x}^{2}+{\hat{p}}_{y}^{2}+{\hat{p}}_{z}^{2})\\ \quad +\,\displaystyle \frac{1}{8}M{{\rm{\Omega }}}^{2}({x}^{2}+{y}^{2}+{z}^{2})+\displaystyle \frac{1}{2}{\omega }_{L}{\hat{L}}_{z},\end{array}\end{eqnarray}$
which is axis-symmetric with respect to the z axis.The initial wave functions are set as the eigenfunctions of angular momentum ${\hat{L}}_{a}$ as [1]
$\begin{eqnarray}{\psi }_{{lm}}^{(a)}({\boldsymbol{x}},t=0)={f}_{l}(r){Y}_{{lm}}^{(a)}(\hat{{\boldsymbol{n}}}),\ (a=x,z),\end{eqnarray}$
where r = ∣x∣, $\hat{{\boldsymbol{n}}}$ is the unit vector of direction in real space. In this paper, we only consider l = 0, 1, 2, which corresponds to s, p, d states. In the coordinates representation, there is $\begin{eqnarray}\begin{array}{rcl}{Y}_{00}^{(z)}({\boldsymbol{x}}) & = & \displaystyle \frac{1}{\sqrt{4\pi }},\\ {Y}_{10}^{(z)}({\boldsymbol{x}}) & = & \sqrt{\displaystyle \frac{3}{4\pi }}\displaystyle \frac{z}{r},\\ {Y}_{1\pm 1}^{(z)}({\boldsymbol{x}}) & = & \sqrt{\displaystyle \frac{3}{8\pi }}\displaystyle \frac{\rho {{\rm{e}}}^{\pm {\rm{i}}\varphi }}{r},\\ {Y}_{20}^{(z)}({\boldsymbol{x}}) & = & \sqrt{\displaystyle \frac{5}{16\pi }}\displaystyle \frac{3{z}^{2}-{r}^{2}}{{r}^{2}},\\ {Y}_{2\pm 1}^{(z)}({\boldsymbol{x}}) & = & \mp \sqrt{\displaystyle \frac{15}{8\pi }}\displaystyle \frac{z\rho {{\rm{e}}}^{\pm {\rm{i}}\varphi }}{{r}^{2}},\\ {Y}_{2\pm 2}^{(z)}({\boldsymbol{x}}) & = & \sqrt{\displaystyle \frac{15}{32\pi }}\displaystyle \frac{{\rho }^{2}{{\rm{e}}}^{\pm 2{\rm{i}}\varphi }}{{r}^{2}},\end{array}\end{eqnarray}$
and $\begin{eqnarray}\begin{array}{rcl}{Y}_{00}^{(x)}({\boldsymbol{x}}) & = & \displaystyle \frac{1}{\sqrt{4\pi }},\\ {Y}_{10}^{(x)}({\boldsymbol{x}}) & = & \sqrt{\displaystyle \frac{3}{4\pi }}\displaystyle \frac{\rho \cos \varphi }{r},\\ {Y}_{1\pm 1}^{(x)}({\boldsymbol{x}}) & = & \sqrt{\displaystyle \frac{3}{8\pi }}\displaystyle \frac{\rho \sin \varphi \pm {\rm{i}}z}{r},\\ {Y}_{20}^{(x)}({\boldsymbol{x}}) & = & \sqrt{\displaystyle \frac{5}{16\pi }}\displaystyle \frac{3{\rho }^{2}{\cos }^{2}\varphi -{r}^{2}}{{r}^{2}},\\ {Y}_{2\pm 1}^{(x)}({\boldsymbol{x}}) & = & \mp \sqrt{\displaystyle \frac{15}{8\pi }}\displaystyle \frac{\rho \cos \varphi (\rho \sin \varphi \pm {\rm{i}}z)}{{r}^{2}},\\ {Y}_{2\pm 2}^{(x)}({\boldsymbol{x}}) & = & \sqrt{\displaystyle \frac{15}{32\pi }}\displaystyle \frac{{\left(\rho \sin \varphi \pm {\rm{i}}z\right)}^{2}}{{r}^{2}},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\rho =\sqrt{{x}^{2}+{y}^{2}},\ \varphi ={\tan }^{-1}\displaystyle \frac{y}{x}.\end{eqnarray}$
To simplify the calculations, we choose $\begin{eqnarray}{f}_{l}(r)={{ \mathcal N }}_{l}{r}^{l}{{\rm{e}}}^{-\tfrac{{r}^{2}}{4\alpha }},\end{eqnarray}$
with ${{ \mathcal N }}_{l}$ the normalization factor. For other forms of initial wave functions, there may be some difference in the evolution of the wave functions. We do not consider these complications in the present paper. It is easy to see that $| {\psi }_{{lm}}^{(z)}({\boldsymbol{x}},0){| }^{2}$ and $| {\psi }_{{lm}}^{(x)}({\boldsymbol{x}},0){| }^{2}$ are axis symmetric with respect to z and x axis, respectively.In the Hamiltonian in equation (4 ), the coupling between the spin momentum and the magnetic field is not included. For a uniform magnetic field, it has no effect on the orbital motion of particles. In the Hamiltonian in equation (4 ), we also have not considered the spin-orbit coupling. The spin-orbit coupling term vanishes for m = 0. For m ≠ 0, usually, it can be approximately neglected when the typical velocity of the particle is less than one percent of the light velocity in a vacuum. For a Gaussian type wave function, it requires the uncertainty in momentum Δpx,y,z/M ≲ 10−2c, where Δpx,y,z are the uncertainty of momenta, and c is the light velocity in vacuum. It implies the profile of the wave function should not be too narrow in real space, and there is α ≳ 104ℏ2/(M2c2). This condition can also be obtained from the explicit expression of the spin-orbit coupling term as shown in [1]. Here we require that the spin-orbit coupling energy is much less than the typical coupling energy between the magnetic field and the magnetic momentum of the particle.
The wave function of the particle evolves as [10]
$\begin{eqnarray}\psi ({\boldsymbol{x}},t)=\int {\rm{d}}{\boldsymbol{x}}^{\prime} K({\boldsymbol{x}},t;{\boldsymbol{x}}^{\prime} ,0)\psi ({\boldsymbol{x}}^{\prime} ,0),\end{eqnarray}$
where the propagator $K({\boldsymbol{x}},t;{\boldsymbol{x}}^{\prime} ,t^{\prime} )$ describes the transition amplitude for a particle progagates from $({\boldsymbol{x}}^{\prime} ,t^{\prime} )$ to (x, t). For ${{\rm{\Omega }}}_{z}=\sqrt{{{\rm{\Omega }}}_{\perp }^{2}+{\omega }_{L}^{2}/4}={\rm{\Omega }}/2$, there is [11] $\begin{eqnarray}\begin{array}{l}K({\boldsymbol{x}},t;{\boldsymbol{x}}^{\prime} ,t^{\prime} )={\left[\displaystyle \frac{M{\rm{\Omega }}}{4\pi {\rm{i}}{\hslash }\sin [{\rm{\Omega }}{\rm{\Delta }}t/2]}\right]}^{\tfrac{3}{2}}\\ \times \,\exp \left\{\displaystyle \frac{{\rm{i}}}{{\hslash }}{ \mathcal A }\right\},\\ { \mathcal A }=\displaystyle \frac{M{\rm{\Omega }}}{2\sin [{\rm{\Omega }}{\rm{\Delta }}t/2]}\left[\displaystyle \frac{1}{2}({x}^{2}+{y}^{2}\right.\\ +\,{x}^{{\prime} 2}+{y}^{{\prime} 2})\cos \displaystyle \frac{{\rm{\Omega }}{\rm{\Delta }}t}{2}\\ -\,({xx}^{\prime} +{yy}^{\prime} )\cos \displaystyle \frac{{\omega }_{L}{\rm{\Delta }}t}{2}\\ +\,({xy}^{\prime} -x^{\prime} y)\sin \displaystyle \frac{{\omega }_{L}{\rm{\Delta }}t}{2}\\ \left.+\,\displaystyle \frac{1}{2}({z}^{2}+z{{\prime} }^{2})\cos \displaystyle \frac{{\rm{\Omega }}{\rm{\Delta }}t}{2}-{zz}^{\prime} \right],\end{array}\end{eqnarray}$
where ${\rm{\Delta }}t=t-t^{\prime} $.3. Results and discussions
By substituting equations (6 ), (7 ) and (11 ) into equation (10 ), we obtain ${\psi }_{{lm}}^{(a)}({\boldsymbol{x}},t),a=x,z$. The results are as the following.
We first obtain the evolution of eigenfunctions of ${\hat{L}}_{z}$. We find that
$\begin{eqnarray}\begin{array}{rcl}{\psi }_{00}^{(z)}({\boldsymbol{x}},t) & = & {{ \mathcal N }}_{0}\displaystyle \frac{1}{\sqrt{4\pi }}{W}^{-\tfrac{3}{2}}g(r),\\ {\psi }_{10}^{(z)}({\boldsymbol{x}},t) & = & {{ \mathcal N }}_{1}\sqrt{\displaystyle \frac{3}{4\pi }}{W}^{-\tfrac{5}{2}}{zg}(r),\\ {\psi }_{1\pm 1}^{(z)}({\boldsymbol{x}},t) & = & {{ \mathcal N }}_{1}\sqrt{\displaystyle \frac{3}{8\pi }}{W}^{-\tfrac{5}{2}}g(r)\rho {{\rm{e}}}^{\pm {\rm{i}}(\varphi -\displaystyle \frac{{\omega }_{L}t}{2})},\\ {\psi }_{20}^{(z)}({\boldsymbol{x}},t) & = & {{ \mathcal N }}_{2}\sqrt{\displaystyle \frac{5}{16\pi }}{W}^{-\tfrac{7}{2}}g(r)({\rho }^{2}-2{z}^{2}),\\ {\psi }_{2\pm 1}^{(z)}({\boldsymbol{x}},t) & = & \mp {{ \mathcal N }}_{2}\sqrt{\displaystyle \frac{15}{32\pi }}{W}^{-\tfrac{7}{2}}g(r)z\rho {{\rm{e}}}^{\pm {\rm{i}}(\varphi -\displaystyle \frac{{\omega }_{L}t}{2})},\\ {\psi }_{2\pm 2}^{(z)}({\boldsymbol{x}},t) & = & {{ \mathcal N }}_{2}\sqrt{\displaystyle \frac{15}{32\pi }}{W}^{-\tfrac{7}{2}}g(r){\rho }^{2}{{\rm{e}}}^{\pm {\rm{i}}2(\varphi -\displaystyle \frac{{\omega }_{L}t}{2})},\end{array}\end{eqnarray}$
where $\begin{eqnarray}\begin{array}{rcl}g(r) & = & \exp \left[-\displaystyle \frac{1}{4\alpha }\displaystyle \frac{2+{\rm{i}}\tfrac{1}{\lambda }(1-{\lambda }^{2})\sin {\rm{\Omega }}t}{1+{\lambda }^{2}+(1-{\lambda }^{2})\cos {\rm{\Omega }}t}{r}^{2}\right],\\ W & = & \cos \displaystyle \frac{{\rm{\Omega }}t}{2}+{\rm{i}}\lambda \sin \displaystyle \frac{{\rm{\Omega }}t}{2},\\ \lambda & = & \displaystyle \frac{{\hslash }}{\alpha M{\rm{\Omega }}}.\end{array}\end{eqnarray}$
We find that the magnetic field induces a phase factor ${{\rm{e}}}^{-{\rm{i}}m{\omega }_{L}t/2}$ to the wave function. This comes from the $\tfrac{1}{2}{\omega }_{L}{\hat{L}}_{z}$ in the Hamiltonian. It is easy to see that the time evolution of the wave function depends on both ωL and Ω. For general values of ωL and Ω, the wave function is not periodical in time. When the ratio ωL/Ω can be written as ratio of two integers, the wave function is periodical in time. Otherwise, the wave function is not a periodical function. This may lead to interesting results in physical phenomena related to the phase of the wave function, such as the Young's double-slit experiment.
It is easy to find that if λ = 1, the probability density $| {\psi }_{{lm}}^{(z)}({\boldsymbol{x}},t){| }^{2}$ is independent of time, while for λ ≠ 1, $| {\psi }_{{lm}}^{(z)}({\boldsymbol{x}},t){| }^{2}$ oscillates with a time period 2π/Ω in the radial direction. Besides, $| {\psi }_{{lm}}^{(z)}({\boldsymbol{x}},t){| }^{2}$ is axis-symmetric with respect to the z axis.
We use the uncertainty in position ${\rm{\Delta }}{{\boldsymbol{x}}}_{i}=\sqrt{\overline{{{\boldsymbol{x}}}_{i}^{2}}-{\overline{{{\boldsymbol{x}}}_{i}}}^{2}},{{\boldsymbol{x}}}_{i}=x,y,z$ to characterize the space distribution of the wave packet. We find that for all the wave functions considered above there is
$\begin{eqnarray}\begin{array}{l}\displaystyle \frac{{\rm{\Delta }}{{\boldsymbol{x}}}_{i}(t)}{{\rm{\Delta }}{{\boldsymbol{x}}}_{i}(t=0)}=\sqrt{\displaystyle \frac{1+{\lambda }^{2}}{2}+\displaystyle \frac{1-{\lambda }^{2}}{2}\cos {\rm{\Omega }}t},\\ {{\boldsymbol{x}}}_{i}=x,y,z.\end{array}\end{eqnarray}$
When λ = 1, Δxi(t) is independent of time. In this case, the initial wave function is actually an eigenfunction of the Hamiltonian. When λ > 1, there is always Δxi(t) ≥ Δxi(t = 0). In this case, the wave function becomes broader due to a weak confining potential. For $\cos {\rm{\Omega }}t=-1$, Δxi(t) takes its maximum value λΔxi(t = 0). For $\cos {\rm{\Omega }}t=1$, Δxi(t) takes its minimum value Δxi(t = 0). When λ < 1, there is Δxi(t) ≤ Δxi(t = 0). In this case, the wave function becomes narrower in real space, due to a strong confining potential. For $\cos {\rm{\Omega }}t=1$, Δxi(t) takes its maximum value Δxi(t = 0). For $\cos {\rm{\Omega }}t=-1$, Δxi(t) takes its minimum value λΔxi(t = 0).We further add comments on the condition to approximately omit the spin-orbit coupling interaction in the Hamiltonian. From the above analysis, we obtain that the approximation requires α(t) ≳ 104ℏ2/(M2c2), where $\alpha (t)=(1+{\lambda }^{2}+(1-{\lambda }^{2})\cos {\rm{\Omega }}t)\alpha $. For λ ≤ 1, the minimum of α(t) is λ2α. So we have λ2α ≳ 104ℏ2/(M2c2). This means that the spin-orbit coupling interaction cannot be neglected when the magnetic field or harmonic potential are too strong. For λ > 1, the minimum value is just α. So we have α ≳ 104ℏ2/(M2c2). This means that for weak magnetic field and harmonic potential, the neglecting of spin-orbit coupling requires the profile of the initial wave packet not be too narrow. For an electron and with λ = 1 it gives ℏΩ ≲ 10−4Mc2 = 5.9 × 105 K. This condition is generally satisfied in atomic physics.
Now we obtain the evolution of eigenfunctions of ${\hat{L}}_{x}$. ${\psi }_{00}^{(x)}({\boldsymbol{x}},t)$ is the same to ${\psi }_{00}^{(z)}({\boldsymbol{x}},t)$. The other wave functions are
$\begin{eqnarray}\begin{array}{l}{\psi }_{10}^{(x)}({\boldsymbol{x}},t)={{ \mathcal N }}_{1}\sqrt{\displaystyle \frac{3}{4\pi }}{W}^{-\tfrac{5}{2}}g(r)\rho \\ \times \,\cos (\varphi -\displaystyle \frac{{\omega }_{L}t}{2}),\\ {\psi }_{1\pm 1}^{(x)}({\boldsymbol{x}},t)={{ \mathcal N }}_{1}\sqrt{\displaystyle \frac{3}{8\pi }}{W}^{-\tfrac{5}{2}}g(r)\\ \times \,\left[\rho \sin (\varphi -\displaystyle \frac{{\omega }_{L}t}{2})\pm {\rm{i}}z\right],\\ {\psi }_{20}^{(x)}({\boldsymbol{x}},t)={{ \mathcal N }}_{2}\sqrt{\displaystyle \frac{5}{16\pi }}{W}^{-\tfrac{7}{2}}g(r)\\ \times \,\left[3{\rho }^{2}{\cos }^{2}(\varphi -\displaystyle \frac{{\omega }_{L}t}{2})-{r}^{2}\right],\\ {\psi }_{2\pm 1}^{(x)}({\boldsymbol{x}},t)=\mp {{ \mathcal N }}_{2}\sqrt{\displaystyle \frac{15}{32\pi }}{W}^{-\tfrac{7}{2}}g(r)\rho \\ \times \,\cos (\varphi -\displaystyle \frac{{\omega }_{L}t}{2})\left[\rho \sin (\varphi -\displaystyle \frac{{\omega }_{L}t}{2})\pm {\rm{i}}z\right],\\ {\psi }_{2\pm 2}^{(x)}({\boldsymbol{x}},t)={{ \mathcal N }}_{2}\sqrt{\displaystyle \frac{15}{32\pi }}{W}^{-\tfrac{7}{2}}g(r)\\ {\times \,\left[\rho \sin (\varphi -\displaystyle \frac{{\omega }_{L}t}{2})\pm {\rm{i}}z\right]}^{2}.\end{array}\end{eqnarray}$
We find that the probability density $| {\psi }_{{lm}}^{(x)}({\boldsymbol{x}},t){| }^{2}$, likes $| {\psi }_{{lm}}^{(z)}({\boldsymbol{x}},t){| }^{2}$, also oscillates in the radial direction with a time period 2π/Ω. The uncertainties in this position satisfy exactly the relation in equation (14 ).
Different from $| {\psi }_{{lm}}^{(z)}{| }^{2}$, here $| {\psi }_{{lm}}^{(x)}{| }^{2}$ (except for $| {\psi }_{00}^{(x)}{| }^{2}$) is anisotropic in the x−y plane, and there exists a symmetry $| {\psi }_{{lm}}^{(x)}(\rho ,z,\varphi ,t){| }^{2}=| {\psi }_{{lm}}^{(x)}(\rho ,z,\varphi +\pi ,t){| }^{2}$. Furthermore, we find that the operator $\tfrac{1}{2}{\omega }_{L}{\hat{L}}_{z}$ in the Hamiltonian makes $| {\psi }_{{lm}}^{(x)}{| }^{2}$ rotating with a time period 2π/ωL with respect to the z axis. For example, there is $| {\psi }_{10}^{(x)}({\boldsymbol{x}},t){| }^{2}\propto {\cos }^{2}(\varphi -\tfrac{{\omega }_{L}t}{2})$ for fixed z. It is anisotropic in the x−y plane with maxima at $\varphi -\tfrac{{\omega }_{L}t}{2}=0,\pi $. As t varies, the value of $| {\psi }_{10}^{(x)}({\boldsymbol{x}},t){| }^{2}$ rotates with a constant angular velocity ωL/2 with respect to the z axis, and the time period is π/(ωL/2) = 2π/ωL.
Since ωL is inverse proportional to the mass of the particle, the particle will probably rotate slowly under a magnetic field. It can be a quasiparticle with a heavy mass, such as the heavy fermions in Kondo lattices [12], fermions in flat bands [13], etc. It can also be a charged atom or molecule. This rotation may be detectable experimentally. The rotation of the anisotropic wave packet probably affects the interactions between atoms or molecules [7, 14].
It is also of interest to consider the case when the rotation is extremely fast, corresponding to a large external magnetic field. In such a case, the profile of the wave packet may be unstable and split into parts, similar to what happened to quantum vortices with quantum numbers too large [15, 16]. Recently, ultracold atom experiments found that when a columnar neutral atomic cluster will be split into smaller parts when it rotates too fast [17].
4. Harmonic oscillating magnetic field
The above results that the probability density of wave functions rotates in a time-independent magnetic field can also be obtained in a phenomenological way using the interaction of the angular momentum of the quantum particle with the magnetic field. There is
$\begin{eqnarray}\displaystyle \frac{{\rm{d}}{\boldsymbol{L}}}{{\rm{d}}t}={\mu }_{{\ell }}{\boldsymbol{L}}\times {\boldsymbol{B}}.\end{eqnarray}$
For an electron, ${\mu }_{{\ell }}=-\tfrac{e}{2{Mc}}$. Since B is along the z direction, the probability density of the eigenfunctions of ${\hat{L}}_{z}$ does not rotate in the magnetic field, while that of ${\hat{L}}_{x}$ rotates along the z axis with an angular velocity proportional to the LxB.Now we extend these results to the situation with an external oscillating magnetic field with ${\boldsymbol{B}}=(0,0,B\cos \omega t)$. Here, we will pay our attention mainly to the rotation of the probability density of the quantum particle under the role of the magnetic field, which is due to the effect of the $\tfrac{1}{2}{\omega }_{L}(t){\hat{L}}_{z}$ term in the Hamiltonian in equation (2 ), where ${\omega }_{L}(t)=\tfrac{{qB}}{{Mc}}\cos \omega t$. Besides, the oscillating magnetic field also contributes to the harmonic potential in the x−y plane. So in the following we only consider the situation with ∣ωL∣/Ω⊥ ≪ 1. Then the contributions due to the oscillating behavior of ωL to the dynamics of the probability density in the radial direction can be treated as a perturbation, and can even be neglected to a good approximation.
From the Maxwell equations, there coexists oscillating electric fields. In the simplest case with propagating electromagnetic fields in which the wavelength is much longer than the typical size of the wave packet of the quantum particle, the vector electric field is perpendicular to the magnetic fields. So to study the rotation of the probability density of the quantum particle, we can omit the effects of electric fields. We just consider the effects of the magnetic fields on the probability density of the quantum particle, which can be described phenomenanically by the interactions between the magnetic fields and the angular momentum.
The $\tfrac{1}{2}{\omega }_{L}(t){\hat{L}}_{z}$ term in the Hamiltonian in equation (2 ) oscillates in time due to the oscillating magnetic field. Similar to that with a constant magnetic field, the probability density of eigenfunction of ${\hat{L}}_{z}$ is not affected by this term. While that for ${\hat{L}}_{x}$, the wave function has the form that the term $\varphi -\tfrac{{\omega }_{L}}{2}t$ in equations (15 ) is substituted with $\varphi -\tfrac{1}{2}{\int }_{0}^{t}{\rm{d}}t^{\prime} \tfrac{{qB}}{{Mc}}\cos \omega t^{\prime} $, which equals $\varphi -\tfrac{{qB}}{2{Mc}\omega }\sin \omega t$. So the probability density of the quantum particle rotates along the z axis with an angular velocity $\tfrac{{qB}}{2{Mc}}\cos \omega t$, which oscillates in time. As a result, the rotation of the probability density oscillates between clockwise and anticlockwise with the magnitude of the angular velocity varying harmonically in time.
5. Conclusions
In this paper, we use a path integral method to calculate the effect of the magnetic field on the evolution of the eigenfunctions of angular momentum. The roles played by the magnetic field on the charged particle are effectively two aspects: a two-dimensional harmonic potential in the plane perpendicular to the magnetic field, which confines the wave packet of the particle; and a term proportional to the orbital angular momentum operator, which rotates wave function around an axis along the direction of the magnetic field. To confine the wave packet in the direction along the magnetic field, an extra harmonic potential is also applied.
In coordinates with the magnetic field along the z direction, we find that whether the initial wave functions are eigenfunctions of ${\hat{L}}_{z}$ or ${\hat{L}}_{x}$, the probability density of the particle oscillates radially with a time period determined by the strength of the effective harmonic potential. We also find that in the case of the initial wave function of ${\hat{L}}_{z}$, the probability density of the particle remains axis-symmetric with respect to the z axis, while the counterpart of ${\hat{L}}_{x}$ is anisotropic in the perpendicular plane, and rotates with a time period inverse proportional to the strength of the magnetic field.
We also extend the results in a phenomenological way to the case with an external magnetic field that varies harmonically in time. We argue that when the effective harmonic potential induced by the magnetic field is much weaker than the external harmonic potential, that is ∣ωL∣/Ω⊥ ≪ 1, the probability density of the eigenfunctions of ${\hat{L}}_{x}$ oscillates between clockwise and anticlockwise with the magnitude of the angular velocity varying harmonically in time.