One Dimensional QM particle Problem

[demanjo]

Dear All, This is my first post. I appreciate your help. I have the following question which i am struggling to understand, let alone solve.

Consider a quantum mechanical particle with the mass m moving in one-dimensional described by the following Hamiltonian;

(1)$$\hat{H} = \frac{\hat{p}^2}{2m} + \frac{K}{2}\hat{x}^6$$
Here, p and x are the operators for the momentum and position, respectively, which satisfy the commutation equation $$[\hat{x},\hat{p}] = i\hbar$$. K is a constant.

1) Express the dimensions of m, hbar, and K in terms of the energy (Joules), the time (seconds) and the length (meter).

2) Express the Energy E₀ and the spatial extent S of the wavefunction for the ground state of the Hamiltonian eq. (1) using u, hbar, and K in terms of the dimensional analysis. Numerical co-efficients are not necessary to be determined.

3) Derive the results of 2) based on the uncertainty principle.
If someone can aid me I would greatly appreciate it.

 

[LiorE]

part 1: This is technical.

In part 2, what is u?

In part 3, think about how to use the uncertainty principle. In the ground state, the kinetic energy and the potential energy can be assumed to be approximately equal. You can also assume that the uncertainty relation holds and that you have (approximately) an equality.

To make use of the uncertainty principle, consider the relation between the kinetic energy and uncertainty in the momentum.

 

[demanjo]

Thank you for the reply.
u is m (mass), sorry, this was a typo.

If i could get through part one, maybe i could have a stab at 2 and 3, considering i am vaguely aware how to use the uncertainty principle, but as you states, 1 is technical, and i cannot even get past that...

 

[demanjo]

No help possible?

 

[LiorE]

Hi,

sorry for the delay.

In q.1 - to get the units, go back to the basics. For example, energy is force times distance, and force is acceleration times mass. From that you can get to the mass. hbar is energy times time, or distance times momentum - you can use those. To get the units of K, note that Kx^6 has the same units has the Hamiltonian. What are the Hamiltonian's units?

 

[demanjo]

Thanks a lot for your reply. I was thinking far too deeply, and didnt think about basics.


My guess would be the Hamiltonian has the units as governed by its components, namely m, and p, where p corresponds to ihbar
So would the units be m/h, which means m/(m^2kg.s) = s/m.kg?