For a free particle in one dimension, calculate the variance...

For a free particle in one dimension, calculate the variance at time $t,(\Delta x)_t^2 \equiv$ $\left\langle\left(x-\langle x\rangle_t\right)^2\right\rangle_t=\left\langle x^2\right\rangle_t-\langle x\rangle_t^2$ without explicit use of the wave function by applying (3.44) repeatedly. Show that $$ (\Delta x)_t^2=(\Delta x)_0^2+\frac{2}{m}\left[\frac{1}{2}\left\langle x p_x+p_x x\right\rangle_0-\langle x\rangle_0\left\langle p_x\right\rangle\right] t+\frac{\left(\Delta p_x\right)^2}{m^2} t^2 $$ and $$ \left(\Delta p_x\right)_t^2=\left(\Delta p_x\right)_0^2=\left(\Delta p_x\right)^2 $$

For a free particle in one dimension, calculate the variance at time $t,(\Delta x)_t^2 \equiv$ $\left\langle\left(x-\langle x\rangle_t\right)^2\right\rangle_t=\left\langle x^2\right\rangle_t-\langle x\rangle_t^2$ without explicit use of the wave function by applying (3.44) repeatedly. Show that
$$
(\Delta x)_t^2=(\Delta x)_0^2+\frac{2}{m}\left[\frac{1}{2}\left\langle x p_x+p_x x\right\rangle_0-\langle x\rangle_0\left\langle p_x\right\rangle\right] t+\frac{\left(\Delta p_x\right)^2}{m^2} t^2
$$
and
$$
\left(\Delta p_x\right)_t^2=\left(\Delta p_x\right)_0^2=\left(\Delta p_x\right)^2
$$

Key Concepts

Expectation Value

In quantum mechanics, the expectation value of an observable is the mean outcome one would obtain from many measurements on an ensemble of identical systems. It is computed as the weighted average of all possible values, where the weights are given by the probability densities. This concept allows one to connect the abstract formalism of operators with measurable quantities, such as position or momentum.

Variance

The variance of an observable quantifies the spread of its measurement outcomes around the expectation value. It is defined as the expectation of the square of the observable minus the square of its expectation value. In the context of quantum mechanics, the variance provides insight into the uncertainty associated with the measurement of the observable, as seen in the uncertainty principle.

Heisenberg Equation of Motion

The Heisenberg equation of motion governs the time evolution of operators in the Heisenberg picture of quantum mechanics. It expresses the time derivative of an operator as proportional to its commutator with the Hamiltonian of the system. This operator-based approach allows one to determine how observables such as position and momentum evolve over time, without explicitly solving for the wave function.

Free Particle Dynamics

For a free particle, the Hamiltonian involves only the kinetic energy term, which is proportional to the square of the momentum operator divided by twice the mass. This results in simple time evolution properties for the observables. The analysis of a free particle highlights the linear time dependence of the position expectation value and the quadratic time dependence of the position variance, while the momentum variance remains constant.

Commutation Relations

The fundamental commutation relations, particularly [x, p] = i?, are central to quantum mechanics. These relations underpin the structure of the theory, influencing the uncertainty relations and the dynamics of observables through their role in the Heisenberg equations of motion. They are essential in deriving the time evolution of statistical properties such as variances.

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