Expectations with respect to the ground state of the harmonic oscillator

1976 ◽  
pp. 80-85
Author(s):  
Sergio A. Albeverio ◽  
Raphael J. Høegh-Krohn
Keyword(s):  
Harmonic Oscillator ◽  
Ground State

GROUND STATE OF N HARMONICALLY BOUND ANYONS IN A MAGNETIC FIELD

1992 ◽  
Vol 07 (38) ◽  
pp. 3593-3600
Author(s):  
R. CHITRA
Keyword(s):  
Magnetic Field ◽  
External Magnetic Field ◽  
Harmonic Oscillator ◽  
Ground State ◽  
Harmonic Frequency ◽  
Oscillator Potential ◽  
Harmonic Oscillator Potential ◽  
Thomas Fermi ◽  
Large N ◽  
The Magnetic Field

The properties of the ground state of N anyons in an external magnetic field and a harmonic oscillator potential are computed in the large-N limit using the Thomas-Fermi approximation. The number of level crossings in the ground state as a function of the harmonic frequency, the strength and the direction of the magnetic field and N are also studied.

Spiked Harmonic Oscillator: N-1 Expansion

2003 ◽  
Vol 17 (14) ◽  
pp. 2761-2772
Author(s):  
S. Datta ◽  
J. K. Bhattacharjee
Keyword(s):  
Harmonic Oscillator ◽  
Ground State ◽  
Numerical Procedure ◽  
Set Up ◽  
Dimensionality Of Space

The ground state of the spiked harmonic oscillator with the potential [Formula: see text] has been obtained variationally for α < 3 and recently by a numerical procedure for α > 3. Due to the Klauder phenomemon at α = 3, analytic techniques do not smoothly interpolate between α < 3 and α > 3. Here we use the N-1 expansion, where N is the dimensionality of space, to set up an analytic scheme that can be continued across α = 3.

Energy bounds for the spiked harmonic oscillator

1995 ◽  
Vol 73 (7-8) ◽  
pp. 493-496 ◽  
Author(s):  
Richard L. Hall ◽  
Nasser Saad
Keyword(s):  
Lower Bound ◽  
Harmonic Oscillator ◽  
Ground State Energy ◽  
Upper Bound ◽  
Ground State ◽  
Trial Function ◽  
State Energy ◽  
Parameter Range ◽  
Complementary Energy ◽  
Energy Bounds

A three-parameter variational trial function is used to determine an upper bound to the ground-state energy of the spiked harmonic-oscillator Hamiltonian [Formula: see text]. The entire parameter range λ > 0 and α ≥ 1 is treated in a single elementary formulation. The method of potential envelopes is also employed to derive a complementary energy lower bound formula valid for all the discrete eigenvalues.

Exact solution of the Faddeev equations for the harmonic oscillator ground state

1980 ◽  
Vol 22 (1) ◽  
pp. 284-286 ◽  
Author(s):  
J. L. Friar ◽  
B. F. Gibson ◽  
G. L. Payne
Keyword(s):  
Exact Solution ◽  
Harmonic Oscillator ◽  
Ground State ◽  
Faddeev Equations

Quantum mechanics of the damped harmonic oscillator

2002 ◽  
Vol 80 (6) ◽  
pp. 645-660 ◽  
Author(s):  
M Blasone ◽  
P Jizba
Keyword(s):  
Harmonic Oscillator ◽  
Ground State ◽  
Geometric Phase ◽  
State Energy ◽  
Dimensional System ◽  
One Dimensional ◽  
Damped Harmonic Oscillator ◽  
Linear Harmonic Oscillator ◽  
Two Dimensional System ◽  
The One

We quantize the system of a damped harmonic oscillator coupled to its time-reversed image, known as Bateman's dual system. By using the Feynman–Hibbs method, the time-dependent quantum states of such a system are constructed entirely in the framework of the classical theory. The geometric phase is calculated and found to be proportional to the ground-state energy of the one-dimensional linear harmonic oscillator to which the two-dimensional system reduces under appropriate constraint. PACS Nos.: 03.65Ta, 03.65Vf, 03.65Ca, 03.65Fd

UNCERTAINTY RELATIONS FOR THE TIME-DEPENDENT SINGULAR OSCILLATOR

2006 ◽  
Vol 20 (10) ◽  
pp. 1211-1231 ◽  
Author(s):  
J. R. CHOI ◽  
I. H. NAHM
Keyword(s):  
Harmonic Oscillator ◽  
Excited States ◽  
Ground State ◽  
Coherent States ◽  
System Approach ◽  
Uncertainty Relation ◽  
Time Dependent ◽  
Uncertainty Relations ◽  
Number State ◽  
State Uncertainty

Uncertainty relations for the time-dependent singular oscillator in the number state and in the coherent state are investigated. We applied our developement to the Caldirola–Kanai oscillator perturbed by a singularity. For this system, the variation (Δx) decreased exponentially while (Δp) increased exponentially with time both in the number and in the coherent states. As k → 0 and χ → 0, the number state uncertainty relation in the ground state becomes 0.583216ℏ which is somewhat larger than that of the standard harmonic oscillator, ℏ/2. On the other hand, the uncertainty relation in all excited states become smaller than that of the standard harmonic oscillator with the same quantum number n. However, as k → ∞ and χ → 0, the uncertainty relations of the system approach the uncertainty relations of the standard harmonic oscillator, (n+1/2)ℏ.

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