Physics

Physics · modern-physics

Problem

Consider a one-dimensional system of Hamiltonian

\hat{H}=\hat{p}^{2} / 2 m+V(\hat{x})

where

V(x)=\lambda x^{n}

. a. Calculate the commutator

[\hat{H}, \hat{x} \hat{p}]

. b. By taking the expectation value of this commutator, show that in any eigenstate of

\hat{H}

, one has the relation:

2\langle T\rangle=n\langle V\rangle

where

\hat{T}=\hat{p}^{2} / 2 m

is the kinetic energy operator. Check this relation on the harmonic oscillator. c. Generalize this result to three dimensions by calculating

[\hat{H}, \hat{\boldsymbol{r}} \cdot \hat{\boldsymbol{p}}]

and considering a potential

V(\boldsymbol{r})

which is a homogeneous function of the variables

x, y, z

, of degree

n

. A homogeneous function of degree

n

satisfies

V(\alpha x, \alpha y, \alpha z)=

\alpha^{n} V(x, y, z)

and

\boldsymbol{r} \cdot \nabla V=n V

. d. Show that for an arbitrary potential

V(r)

one has the general relation:

2\langle T\rangle=\left\langle r \frac{\partial V}{\partial r}\right\rangle