- #1
demanjo
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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)[tex]\hat{H} = \frac{\hat{p}^2}{2m} + \frac{K}{2}\hat{x}^6 [/tex]
Here, p and x are the operators for the momentum and position, respectively, which satisfy the commutation equation [tex][\hat{x},\hat{p}] = i\hbar[/tex]. 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.
Consider a quantum mechanical particle with the mass m moving in one-dimensional described by the following Hamiltonian;
(1)[tex]\hat{H} = \frac{\hat{p}^2}{2m} + \frac{K}{2}\hat{x}^6 [/tex]
Here, p and x are the operators for the momentum and position, respectively, which satisfy the commutation equation [tex][\hat{x},\hat{p}] = i\hbar[/tex]. 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.