Uncertainty Product of Wave Packet in Harmonic Potential

Jürgen Brickmann, Peter Russegger

Sang Pyo Kim, Kyoung Kon Kim

Daniel M. Dubois, Daniel M. Dubois

Joydev Lahiri, V Jayaram Menon

The problem of finding the class of wave packets \ensuremath{\psi} that minimizes the quantum-mechanical uncertainty product ${\ensuremath{\Delta}}_{x}$${\ensuremath{\Delta}}_{p}$ is treated via the constrained variational principle. This leads to a harmonic-oscillator-type wave equation for \ensuremath{\psi} whose explicit solutions are obtained when the potential V has, or does not have, an infinite step. The method can be fruitfully applied to the quantum cosmology of the inflationary universe.

Naohisa Ogawa, Shuichi Nagasawa

I. A. Pedrosa, B. F. Ramos, K. Bakke +1 more

We discuss the quantum theory of an harmonic oscillator with time-dependent mass and frequency submitted to action of a complex time-dependent linear potential with [Formula: see text] symmetry. Combining the Lewis and Riesenfeld approach to time-dependent non-Hermitian Hamiltonians having [Formula: see text] symmetry and linear invariants, we solve the time-dependent Schrödinger equation for this problem and use the corresponding quantum states to construct a Gaussian wave packet solution. We show that the shape of this wave packet does not depend on the driving force. Afterwards, using this wave packet state, we calculate the expectation values of the position and momentum, their fluctuations and the associated uncertainty product. We find that these expectation values are complex numbers and as a consequence the position and momentum operators are not physical observables and the uncertainty product is physically unacceptable.