Question

In the Heisenberg picture, operators evolve according to

∂tA=1iℏ[A,H].

My question is, does the following relation hold?

(X_H)²=(X²)H

The system (Hamiltonian) in mind is the 1D harmonic oscillator,

H=a^†a+12

I would like to say that it does, but the Heisenberg equations of motion for x² and p² are very messy (and I believe nonlinear).

Answer 1

The relation
`(X_H)^2=(X^2)_H` does not generally hold in the Heisenberg picture of quantum mechanics, particularly for the harmonic oscillator, as the Heisenberg evolution of operators can be quite complex due to nonlinearity and commutation with the Hamiltonian. The Heisenberg uncertainty principle is a separate concept that limits the precision of simultaneous measurement of conjugate quantities like position and momentum.

Step-by-step explanation:

The question you've asked pertains to the Heisenberg picture in quantum mechanics, specifically how operators evolve over time within this framework. In the Heisenberg picture, it is indeed true that operators evolve according to the equation
`\partial tA=1/i\hbar [A,H]`, where A is an observable, H is the Hamiltonian of the system, and [A,H] denotes the commutator of A and H. When it comes to the 1D harmonic oscillator, the Hamiltonian is given by H=a†a+1/2, with a† being the creation operator and a the annihilation operator.

However, the relationship
`(X_H)^2=(X^2)_H` does not hold in general because the squared position operator in the Heisenberg picture,
`(X_H)^2`, can involve time evolution and commutators with the Hamiltonian that do not simplify to the Heisenberg picture evolution of X².

The Heisenberg equations of motion for x² and p² can indeed become quite complicated due to their nonlinearity. In the context of quantum mechanics and specifically for the harmonic oscillator, the calculations of such expressions require careful treatment that involves commutation relations and possibly perturbation methods or numerical solutions.

The Heisenberg uncertainty principle, while not directly related to the evolution of operators, is fundamentally about the limits of measuring certain pairs of physical quantities, such as position and momentum, simultaneously. It asserts that the more precisely one quantity is known, the less precisely the conjugate quantity can be known, and this is represented mathematically as ΔxΔp≥ h/4π.