Position and momentum uncertainties of the normal and inverted harmonic oscillators under the minimal length uncertainty relation

We analyze the position and momentum uncertainties of the energy eigenstates of the harmonic oscillator in the context of a deformed quantum mechanics, namely, that in which the commutator between the position and momentum operators is given by $[\widehat{x},\widehat{p}]=i\hslash (1+\beta {\widehat{p}}^2)$. This deformed commutation relation leads to the minimal length uncertainty relation $\Delta x\ge (\hslash /2)(1/\Delta p+\beta \Delta p)$, which implies that $\Delta x\sim 1/\Delta p$ at small $\Delta p$ while $\Delta x\sim \Delta p$ at large $\Delta p$. We find that the uncertainties of the energy eigenstates of the normal harmonic oscillator ($m>0$), derived in L. N. Chang, D. Minic, N. Okamura, and T. Takeuchi, Phys. Rev. D 65, 125027 (2002), only populate the $\Delta x\sim 1/\Delta p$ branch. The other branch, $\Delta x\sim \Delta p$, is found to be populated by the energy eigenstates of the “inverted” harmonic oscillator ($m<0$). The Hilbert space in the inverted case admits an infinite ladder of positive energy eigenstates provided that $\Delta x_{\min }=\hslash \sqrt{\beta }>\sqrt{2}[{\hslash }^2/k|m|{]}^{1/4}$. Correspondence with the classical limit is also discussed.

Figure 1

The solid line indicates the lower bound on $\Delta x$ as a function of $\Delta p$ when they obey the minimal length uncertainty relation, Eq. (2). The dashed lines indicate $\Delta x=(\hslash /2)(1/\Delta p)$ and $\Delta x=(\hslash \beta /2)\Delta p$.

Figure 2

Plots of ${\lambda }_{+}$ and ${\lambda }_{-}$, Eq. (23), as functions of ${\kappa }^2=\beta \hslash \sqrt{k|m|}$.

Figure 3

$\lambda$-dependence of the wave functions of the first few energy eigenstates. The $\lambda >1$ values correspond to $m>0$, while the $\frac{1}{2}\le \lambda <1$ values correspond to $m<0$. The $\lambda =1$ case corresponds to the limit $|m|\to \infty$.

Figure 4

$\Delta p$ versus $\Delta x$ for the lowest six energy eigenstates of the harmonic oscillator. $\Delta x$ is in units of $\Delta x_{\min }=\hslash \sqrt{\beta }$, while $\Delta p$ is in units of $1/\sqrt{\beta }=\hslash /\Delta x_{\min }$. The location of the state along the curves shown is determined by the value of $\lambda$ defined in Eq. (23). The $\lambda =1$ points correspond to the case $1/m=0$. As $1/m$ is increased to the positive side, the value of $\lambda$ will increase away from one and the state will move toward the left along the $\Delta x\sim 1/\Delta p$ branch. If $1/m$ is decreased into the negative, the value of $\lambda$ will decrease toward $1/2$, and the state will move toward the right along the $\Delta x\sim \Delta p$ branch. The $n=0$ curve corresponds to the minimal length uncertainty bound, Eq. (2).

Figure 5

The dependence of the classical behavior of a positive-mass particle in a harmonic oscillator potential on the parameter $C=2mE\beta =\beta m^2{\omega }^2{A}^2$, where $E$ is the particle’s energy, and $A$ is the amplitude of the oscillation in $x$. The dashed line indicates the undeformed $\beta =0$ case, while the four solid lines indicate the $C=1$, 2, 4, and 8 cases with larger values of $C$ leading to shorter oscillation periods.

Figure 6

The classical behavior of a positive-mass particle in a harmonic oscillator potential with modified Poisson brackets, Eq. (55), in the limit $m\to \infty$ with $C=2mE\beta$ where $E$ and $\beta$ are kept fixed. $\rho (t)$ and $x(t)$ take on the behavior of the position and momentum of a particle in an infinite square well.

Figure 7

The dependence of the classical behavior of a negative-mass particle in a harmonic oscillator potential on the parameter $C=-2\mu E\beta =-\beta |m|k{B}^2$, where $E$ is the particle’s energy, and $B$ is the distance of closest approach to the origin in $x$ space. Note that, due to the negative mass, $p(t)$ is negative when $\dot{x}(t)$ is positive, and vice versa. The dashed line indicates the undeformed $\beta =0$ case. The four solid lines indicate the $C=-\frac{1}{16}$, $-\frac{1}{4}$, $-1$, and $-4$ cases, with larger values of $|C|$ leading to shorter travel times from $x=-\infty$ to $x=-B$ and back.

Figure 8

The dependence of the classical behavior of a negative-mass particle in a harmonic oscillator potential on the parameter $C=-2|m|E\beta =\beta p_{\min }^2$, where $E$ is the particle’s energy, and $p_{\min }$ is the momentum of the particle at the origin $x=0$. The dashed line indicates the undeformed $\beta =0$ case. The four solid lines indicate the $C=+\frac{1}{16}$, $+\frac{1}{4}$, $+1$, and $C=+4$ cases with larger values of $C$ leading to shorter travel times from $x=-\infty$ to $x=+\infty$.

Figure 9

$\omega T$ as a function of $-C=2|m|E\beta$, where $\omega =\sqrt{k/|m|}$, and $E$ is the particle’s energy. $T/2$ is the time it takes for the particle to traverse the entire trajectory.

Figure 10

The periodic behavior of the negative-mass particle in a harmonic oscillator potential in compactified $x$ space. The example shown is for $C=-1$.

Figure 11

Comparison of quantum and classical probabilities in $\rho$, $p$, and $x$ spaces for a negative-mass particle in a harmonic oscillator potential for the case $\lambda =\frac{3}{4}$ and $n=30$, which corresponds to the classical $C=-2|m|E\beta =-5044$. The quantum distribution in $x$ space is discrete due to the existence of the minimal length. There is also some seepage of the probability into classically forbidden regions in $x$ space as is expected of quantum probabilities.