If \(H=p^{2} / 2 \mu+V(x)\) for one-dimensional motion of a particle, and \(V(x)\) can be expressed as a power series in \(x\), show by purely matrix methods that $$ \frac{d x}{d t}=\frac{p}{\mu} \quad \frac{d p}{d t}=-\frac{d V}{d x} $$ where \(x\) and \(p\) are in the Heisenberg picture.

In quantum mechanics, the Hamiltonian represents the total energy of a system. It's a central concept because it dictates how a quantum system evolves over time. For a particle moving in one dimension, the Hamiltonian can be expressed as:

\[ H = \frac{p^2}{2\mu} + V(x) \]

This formula combines the kinetic energy term \(\frac{p^2}{2\mu}\) and the potential energy term \(V(x)\). Here, \(p\) is the momentum, \(\mu\) is the mass, and \(x\) represents the position.

• The kinetic energy term comes from the motion of the particle.
• The potential energy term depends on the position and arises from forces acting on the particle.

Understanding the Hamiltonian helps to explore how quantum states change continuously over time.

The commutator is a mathematical concept used to measure the difference between two operations performed in succession. For operators \(A\) and \(B\), it is defined as:

\[ [A, B] = AB - BA \]

This concept is essential in quantum mechanics to understand how quantities like position and momentum interact. For example, the commutator of the Hamiltonian \(H\) and the position operator \(x\) in the Heisenberg picture is used to find the time evolution of \(x\):

\[ \left[ H, x \right] = \left[ \frac{p^2}{2\mu} + V(x), x \right] \]

Evaluating this commutator, considering the properties of momentum \(p\) and potential energy \(V(x)\), allows us to derive important time evolution equations, showing the movement and interaction of a particle in a potential field.

The property:

• \( [p, x] = -i\hbar \)

Is particularly crucial since it ties directly into how the fundamental observables of quantum systems are defined and operate.

In quantum mechanics, operators can evolve over time, especially in the Heisenberg picture. To find the time evolution of an operator \(O\), we use the equation:

\[ \frac{dO}{dt} = \frac{i}{\hbar} [H, O] + \left(\frac{\partial O}{\partial t}\right)_H \]

For many cases, including those of position \(x\) and momentum \(p\), the operators do not have explicit time dependence, so the partial derivative term \(\left(\frac{\partial O}{\partial t}\right)_H\) vanishes. This simplifies our equation to:

• \( \frac{dO}{dt} = \frac{i}{\hbar} [H, O] \)

For instance, the time evolution of the position operator \(x\) is given by:

\[ \frac{dx}{dt} = \frac{i}{\hbar} [H, x] \]

And for the momentum operator \(p\):

\[ \frac{dp}{dt} = \frac{i}{\hbar} [H, p] \]

Solving these equations, with the proper commutators calculated, reveals the dynamics of the system. As shown for our particle, we get:

• \( \frac{dx}{dt} = \frac{p}{\mu} \)
• \( \frac{dp}{dt} = -\frac{dV}{dx} \)

These results illustrate the rules governing the motion of the particle, reminiscent of classical mechanics but derived from quantum principles.