The Heisenberg Picture: Equations of Motion for Operators

Quantum Mechanics A

Schrödinger Equation The most fundamental equation of quantum mechanics; given a Hamiltonian Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}} , it describes how a state Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi\rangle} evolves in time.

Physical Basis of Quantum Mechanics

Basic Concepts and Theory of Motion UV Catastrophe (Black-Body Radiation) Photoelectric Effect Stability of Matter Double Slit Experiment Stern-Gerlach Experiment The Principle of Complementarity The Correspondence Principle The Philosophy of Quantum Theory

Schrödinger Equation

Brief Derivation of Schrödinger Equation Relation Between the Wave Function and Probability Density Stationary States Heisenberg Uncertainty Principle Some Consequences of the Uncertainty Principle

Operators, Eigenfunctions, and Symmetry

Linear Vector Spaces and Operators Commutation Relations and Simultaneous Eigenvalues The Schrödinger Equation in Dirac Notation Transformations of Operators and Symmetry Time Evolution of Expectation Values and Ehrenfest's Theorem

Motion in One Dimension

One-Dimensional Bound States Oscillation Theorem The Dirac Delta Function Potential Scattering States, Transmission and Reflection Motion in a Periodic Potential Summary of One-Dimensional Systems

Discrete Eigenvalues and Bound States

Harmonic Oscillator Spectrum and Eigenstates Analytical Method for Solving the Simple Harmonic Oscillator Coherent States Charged Particles in an Electromagnetic Field WKB Approximation

Time Evolution and the Pictures of Quantum Mechanics

The Heisenberg Picture: Equations of Motion for Operators The Interaction Picture The Virial Theorem

Angular Momentum

Commutation Relations Angular Momentum as a Generator of Rotations in 3D Spherical Coordinates Eigenvalue Quantization Orbital Angular Momentum Eigenfunctions

Central Forces

General Formalism Free Particle in Spherical Coordinates Spherical Well Isotropic Harmonic Oscillator Hydrogen Atom WKB in Spherical Coordinates

The Path Integral Formulation of Quantum Mechanics

Feynman Path Integrals The Free-Particle Propagator Propagator for the Harmonic Oscillator

Continuous Eigenvalues and Collision Theory

Differential Cross Section and the Green's Function Formulation of Scattering Central Potential Scattering and Phase Shifts Coulomb Potential Scattering

There are several ways to mathematically approach the change of a quantum mechanical system with time. The Schrödinger picture considers wave functions which change with time while the Heisenberg picture places the time dependence on the operators and deals with wave functions that do not change in time. The interaction picture, also known as the Dirac picture, is somewhere between the other two, placing time dependence on both operators and wave functions. The two "pictures" correspond, theoretically, to the time evolution of two different things. Mathematically, all methods should produce the same result.

Definition of the Heisenberg Picture

The time evolution operator, defined as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi(t)\rangle=\hat{U}(t,t_0)|\Psi(t_0)\rangle,}

is at the heart of the "evolution" of a state, and is the same in each picture. The evolution operator is obtained from the time dependent Schrödinger equation as follows. We start by rewriting the equation as

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\hbar \frac{d}{dt}\hat{U}(t,t_0)|\Psi(t_0)\rangle = \hat{H}\hat{U}(t,t_0)|\Psi(t_0)\rangle. }

Since this equation will hold regardless of our choice of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi(t_0)\rangle,} we conclude that

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\hbar \frac{d}{dt}\hat{U}(t,t_0)=\hat{H}\hat{U}(t,t_0). }

It follows from the fact that the Schrödinger equation preserves the normalization of the state vector for all times that the time evolution operator is unitary:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{U}^\dagger(t,t_0)\hat{U}(t,t_0)=\hat{I}}

If we know the time evolution operator, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{U},} and the initial state of a particular system, all that is needed to determine the state at a later time is to apply the time evolution operator to the initial state vector:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{U}(t,t_0)|\Psi(t_0)\rangle=|\Psi(t)\rangle. }

Therefore, if we know the initial state of a system, we can also obtain the expectation value of an operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} at some later time:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle\hat{A}\rangle(t) = \langle\Psi(t)|\hat{A}|\Psi(t)\rangle= \langle \Psi(t_0)|\hat{U}^{\dagger}(t,t_0)\hat{A}\hat{U}(t,t_0)|\Psi(t_0)\rangle}

We may view the above equation in two ways. One way is to think of the operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} as time-independent and to consider all of the time dependence of its expectation value as coming from the state vector. This point of view is known as the Schrödinger picture. Alternatively, we may think of the operator as evolving in time, while the state vector stays constant. This is known as the Heisenberg picture. In this picture, the time evolution of the operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}} is given by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{A}_H(t)=\hat{U}^{\dagger}(t,t_0)\hat{A}\hat{U}(t,t_0)}

and our state vector is just the initial state vector Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi(t_0)\rangle.} Note that the only difference between the two pictures is in the way that we assign the time dependence of the system; i.e., whether we think of the state as evolving or of the operators as evolving.

The Heisenberg picture is useful because we can see a closer connection to classical physics than with the Schrödinger picture. In classical physics, we describe the evolution of a system in terms of the time evolution of the observables, such as position or angular momentum, as dictated by the classical equations of motion. Classical mechanics does not include the concept of state vectors, as quantum mechanics does.

Comparing the Heisenberg Picture and the Schrödinger Picture

As mentioned above, both the Heisenberg representation and the Schrödinger representation give the same results for the time dependent expectation values of operators.

In the Schrödinger picture, in which we are most accustomed, the states change over time while the operators remain constant. In other words, the time dependence is carried by the state operators. When the Hamiltonian is independent of time, it is possible to write:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi(0)\rangle }

as the state at t = 0.

At another time t, this becomes

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle |\Psi(t)\rangle = e^{-iHt/\hbar} |\Psi(0)\rangle } ,

which in this form solves the Schrödinger equation

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\hbar\frac{\partial|\Psi(t)\rangle}{\partial t}=H|\Psi(t)\rangle } .

Conversely, in the Heisenberg picture to find the time dependent expectation of a given operator, the states remain constant while the operators change in time. In this case, you can write the time dependent operator as:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A(t) = e^{iHt/\hbar}Ae^{-iHt\hbar} } ,

which means the expectation value is,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \langle A \rangle_t = \langle\Psi(0)| A(t)| \Psi(0)\rangle } .

The Heisenberg Equation of Motion

In the Heisenberg picture, the quantum mechanical observables change in time as dictated by the Heisenberg equations of motion. We can study the evolution of a Heisenberg operator by differentiating equation 2 with respect to time:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{d}{dt} A_H &= \frac{\partial U^{\dagger}}{\partial t} A U + U^{\dagger}\frac{\partial A}{\partial t} U + U^{\dagger} A \frac{\partial U}{\partial t} \\ &= \frac{-1}{i\hbar} U^{\dagger} HUU^{\dagger}A U + \left(\frac{\partial A}{\partial t}\right)_H + U^{\dagger} AUU^{\dagger}HU \frac{1}{i\hbar} \\ &= \frac{1}{i\hbar}\left(\left[A,H\right]\right)_H+\left(\frac{\partial A}{\partial t}\right)_H. \end{align} }

The last equation is known as the Ehrenfest's Theorem.

For example, if we have a hamiltonian of the form,

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H=\frac{p^2}{2m}+V(r),}

then we can find the Heisenberg equations of motion for p and r.

The position operator in 3D is: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i\hbar\frac{d \hat{r}(t)}{dt} = \left[ \hat{r}(t),H\right] }

Since the Hamiltonian as given above has a V(r,t) term and a momentum term the two commuators are as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ \hat{r}(t),V(r,t)\right] = 0 }

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ \hat{r}(t), \frac{p(t)^{2}}{2m}\right] = \frac{i\hbar p(t)}{m} }

this yields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d \hat{r}(t)_{H}}{dt} = \frac{ \hat{p}(t)_{H}}{m} } .

To find the equations of motion for the momentum you need to evaluate Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ \hat{p}(t), V(r(t))\right] } ,

which equals, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[\hat{p}, V(r)\right] = -i\hbar \nabla V(r) } .

This yields Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d \hat{p}_H}{dt}=-\nabla V(\hat{r}_H).}

These results are, of course, what we would expect if we only knew classical physics, providing another example of how the Heisenberg picture is more transparently similar to classical physics.

In particular, if we apply these equations to the Harmonic oscillator with natural frequency Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \omega=\sqrt{\frac{k}{m}}\!}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d\hat x_H }{dt}=\frac{\hat p_H }{m}}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{d\hat p_H }{dt}=-k{\hat x_H}. }

we can solve the above equations of motion and find

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat x_H(t)=\hat x_H(0)\cos(\omega t)+\frac{\hat p_H(0)}{m\omega}\sin(\omega t)}

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat p_H(t)=\hat p_H(0)\cos(\omega t)-\hat x_H(0)m\omega\sin(\omega t).\!}

It is important to stress that the above oscillatory solution is for the position and momentum operators. Also, note that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat x_H(0)\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat p_H(0)\!} correspond to the time independent operators Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat x\!} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat p\!} .

Motion in electromagnetic field

The Hamiltonian of a particle of charge Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle e\!} and mass Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m\!} in an external electromagnetic field, which may be time-dependent, is given as follows:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H=\frac{1}{2m}\left(\mathbf{p}-\frac{e}{c}\bold A(\bold r,t)\right)^2+e\phi(\bold r,t)}

where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold{A(\bold r,t)} \!} is the vector potential and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\phi(\bold r,t)}\!} is the Coulomb potential of the electromagnetic field. In a problem, if there is a momentum operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold p\!} , it must be replaced by

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right)}

if a particle is under the influence of an electromagnetic field.

Let's find out the Heisenberg equations of motion for the position and velocity operators. For position operatorFailed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold r\!} , we have:

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \begin{align} \frac{d\bold r}{dt} &= \frac{1}{i\hbar} \left[\bold r,H \right] \\ &= \frac{1}{i\hbar} \left[ \bold r, \frac{1}{2m} \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right)^2 + e\phi(\bold r,t)\right] \\ &= \frac{1}{2im\hbar} \left[\bold r, \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right)^2\right] \\ &= \frac{1}{2im\hbar} \left[\bold r, \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right)\right]\left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right) + \frac{1}{2im\hbar} \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right) \left[\bold r, \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right)\right] \\ &= \frac{1}{2im\hbar} \left[\bold r, \bold p\right] \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right) + \frac{1}{2im\hbar} \left(\bold p - \frac{e}{c}\bold A(\bold r,t)\right) \left[\bold r, \bold p\right] \\ &= \frac{1}{2im\hbar}i\hbar \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right) + \frac{1}{2im\hbar} \left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right)i\hbar \\ &= \frac{1}{m}\left(\bold p-\frac{e}{c}\bold A(\bold r,t)\right), \end{align} }

where (Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold r \!} does not depend on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t \!} explicitly) is the equation of motion for the position operator Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \bold r} . This equation also defines the velocity operator ${\mathbf {v}}$:

${\mathbf {v}}={\frac {1}{m}}\left({\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t)\right)$

The Hamiltonian can be rewritten as:

$H={\frac {m}{2}}{\mathbf {v}}\cdot {\mathbf {v}}+e\phi$

Therefore, the Heisenberg equation of motion for the velocity operator is:

${\begin{aligned}{\frac {d{\mathbf {v}}}{dt}}&={\frac {1}{i\hbar }}\left[{\mathbf {v}},H\right]+{\frac {\partial {\mathbf {v}}}{\partial t}}\\&={\frac {1}{i\hbar }}\left[{\mathbf {v}},{\frac {m}{2}}{\mathbf {v}}\cdot {\mathbf {v}}\right]+{\frac {1}{i\hbar }}\left[{\mathbf {v}},e\phi \right]-{\frac {e}{mc}}{\frac {\partial {\mathbf {A}}}{\partial t}}\end{aligned}}$

(Note that ${\mathbf {p}}\!$ does not depend on $t\!$ expicitly)

Let's use the following commutator identity:

$\left[{\mathbf {v}},{\mathbf {v}}\cdot {\mathbf {v}}\right]={\mathbf {v}}\times \left({\mathbf {v}}\times {\mathbf {v}}\right)-\left({\mathbf {v}}\times {\mathbf {v}}\right)\times {\mathbf {v}}$

Substituting, we get:

${\frac {d{\mathbf {v}}}{dt}}={\frac {1}{i\hbar }}{\frac {m}{2}}\left({\mathbf {v}}\times ({\mathbf {v}}\times {\mathbf {v}})-({\mathbf {v}}\times {\mathbf {v}})\times {\mathbf {v}}\right)+{\frac {1}{i\hbar }}e[{\mathbf {v}},\phi ]-{\frac {e}{mc}}{\frac {\partial {\mathbf {A}}}{\partial t}}$

Now let's evaluate ${\mathbf {v}}\times {\mathbf {v}}\!$ and $[{\mathbf {v}},\phi ]\!$:

${\begin{aligned}({\mathbf {v}}\times {\mathbf {v}})_{i}&=\epsilon _{ijk}v_{j}v_{k}\\&=\epsilon _{ijk}{\frac {1}{m}}\left(p_{j}-{\frac {e}{c}}A_{j}({\mathbf {r}},t)\right){\frac {1}{m}}\left(p_{k}-{\frac {e}{c}}A_{k}({\mathbf {r}},t)\right)\\&=-{\frac {e}{m^{2}c}}\epsilon _{ijk}\left(p_{j}A_{k}({\mathbf {r}},t)+A_{j}({\mathbf {r}},t)p_{k}\right)\\&=-{\frac {e}{m^{2}c}}\epsilon _{ijk}p_{j}A_{k}({\mathbf {r}},t)-{\frac {e}{m^{2}c}}\epsilon _{ijk}A_{j}({\mathbf {r}},t)p_{k}\\&=-{\frac {e}{m^{2}c}}\epsilon _{ijk}p_{j}A_{k}({\mathbf {r}},t)-{\frac {e}{m^{2}c}}\epsilon _{ikj}A_{k}({\mathbf {r}},t)p_{j}{\mbox{(Switching indices in the second terms)}}\\&=-{\frac {e}{m^{2}c}}\epsilon _{ijk}p_{j}A_{k}({\mathbf {r}},t)+{\frac {e}{m^{2}c}}\epsilon _{ijk}A_{k}({\mathbf {r}},t)p_{j}\\&=-{\frac {e}{m^{2}c}}\epsilon _{ijk}\left[p_{j},A_{k}({\mathbf {r}},t)\right]\\&=-{\frac {e}{m^{2}c}}\epsilon _{ijk}{\frac {\hbar }{i}}\nabla _{j}A_{k}({\mathbf {r}},t)\\&=i\hbar {\frac {e}{m^{2}c}}\left(\nabla \times {\mathbf {A}}\right)_{i}\end{aligned}}$

$\rightarrow \left[{\mathbf {v}}\times {\mathbf {v}}\right]=i\hbar {\frac {e}{m^{2}c}}\left(\nabla \times {\mathbf {A}}\right)=i\hbar {\frac {e}{m^{2}c}}{\mathbf {B}}$

and

${\begin{aligned}\left[{\mathbf {v}},\phi \right]&={\frac {1}{m}}\left[{\mathbf {p}}-{\frac {e}{c}}{\mathbf {A}}({\mathbf {r}},t),\phi ({\mathbf {r}},t)\right]\\&={\frac {1}{m}}\left[{\mathbf {p}},\phi ({\mathbf {r}},t)\right]\\&={\frac {1}{m}}{\frac {\hbar }{i}}\nabla \phi \end{aligned}}$

Substituting and rearranging, we get:

$m{\frac {d{\mathbf {v}}}{dt}}={\frac {e}{2c}}\left({\mathbf {v}}\times {\mathbf {B}}-{\mathbf {B}}\times {\mathbf {v}}\right)+e{\mathbf {E}}$

where

${\mathbf {E}}=-\nabla \phi -{\frac {1}{c}}{\frac {\partial {\mathbf {A}}}{\partial t}}$

Above is the quantum mechanical version of the equation for the acceleration of the particle in terms of the Lorentz force.

These results can also be deduced in Hamiltonian dynamics due to the similarity between the Hamiltonian dynamics and quantum mechanics.