Position and momentum uncertainties of a particle in a \(V\)-shaped potential under the minimal length uncertainty relation. (English) Zbl 1337.81064
Summary: We calculate the uncertainties in the position and momentum of a particle in the 1D potential \(V(x)=F|x|\), \(F>0\), when the position and momentum operators obey the deformed commutation relation \([\hat x,\hat p]=i\hbar(1+\beta\hat p^2)\), \(\beta>0\). As in the harmonic oscillator case, which was investigated in a previous publication, the Hamiltonian \(\hat H_1=\hat p^2/2m+F|\hat x|\) admits discrete positive energy eigenstates for both positive and negative mass. The uncertainties for the positive mass states behave as \(\Delta x\sim 1/\Delta p\) as in the \(\beta=0\) limit. For the negative mass states, however, in contrast to the harmonic oscillator case where we had \(\Delta x\sim\Delta p\), both \(\Delta x\) and \(\Delta p\) diverge. We argue that the existence of the negative mass states and the divergence of their uncertainties can be understood by taking the classical limit of the theory. Comparison of our results is made with previous work by S. Z. Benczik [Investigations on the minimal-length uncertainty relation. Ph.D. Thesis, Virginia Tech (2007)].
MSC:
| 81Q35 | Quantum mechanics on special spaces: manifolds, fractals, graphs, lattices |
| 81S05 | Commutation relations and statistics as related to quantum mechanics (general) |
| 83C45 | Quantization of the gravitational field |
| 81Q10 | Selfadjoint operator theory in quantum theory, including spectral analysis |
Keywords:
minimal length uncertainty relation; generalized uncertainty relation; position and momentum uncertaintiesReferences:
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