A comparison of classical and quantum uncertainties is presented for the particle-in-a-box, the harmonic oscillator, and the one-electron atom. It is found that the quantum results reduce to the classical in the limit of either very large quantum numbers or h→0. A classical uncertainty principle is derived and compared with its quantum analogue. A possible relationship between zero-point motion and the uncertainty principle is noted.
A comparison of classical and quantum uncertainties is presented for the particle-in-a-box, the harmonic oscillator, and the one-electron atom. It is found that the quantum results reduce to the classical in the limit of either very large quantum numbers or h→0. A classical uncertainty principle is derived and compared with its quantum analogue. A possible relationship between zero-point motion and the uncertainty principle is noted.
Abstract Heisenberg’s uncertainty principle is often cited as an example of a ‘purely quantum’ relation with no analogue in the classical limit where ℏ → 0. However, this formulation of the classical limit is problematic for many reasons, one of which is dimensional analysis. Since ℏ is a dimensionful constant, we may always work in natural units in which ℏ = 1. Dimensional analysis teaches us that all physical laws can be expressed purely in terms of dimensionless quantities. This indicates that the existence of a dimensionally consistent constraint on ΔxΔp requires the existence of a dimensionful parameter with units of action, and that any definition of the classical limit must be formulated in terms of dimensionless quantities (such as quantum numbers). Therefore, bounds on classical uncertainty (formulated in terms of statistical ensembles) can only be written in terms of dimensionful scales of the system under consideration, and can be readily compared to their quantum counterparts after being non-dimensionalized. We compare the uncertainty of certain coupled classical systems and their quantum counterparts (such as harmonic oscillators and particles in a box), and show that they converge in the classical limit. We find that since these systems feature additional dimensionful scales, the uncertainty bounds are dependent on multiple dimensionless parameters, in accordance with dimensional considerations.
Abstract The correspondence principle provides a prescription to connect quantum physics to classical. It asserts that the physical quantities evaluated quantum mechanically approach their respective classical values for large quantum numbers. This has been shown for the pedagogically important cases of the particle in a box and a harmonic oscillator. However, a particle in a constant field has a wave function related to the Airy function and has at best been treated numerically. Employing energy eigenstates we obtain the expectation values of the position, the momentum and their moments upto fourth order, rigorously and without resorting to numerical or graphical techniques. We compare them with the corresponding classical values. We also examine the uncertainty product for the system.
Abstract As an undergraduate exercise, in an article (2012 Am. J. Phys. 80 780–14 ), quantum and classical uncertainties for dimensionless variables of position and momentum were evaluated in three potentials: infinite well, bouncing ball, and harmonic oscillator. While original quantum uncertainty products depend on ℏ and the number of states ( n ), a dimensionless approach makes the comparison between quantum uncertainty and classical dispersion possible by excluding ℏ . But the question is whether the uncertainty still remains dependent on quantum number n . In the above-mentioned article, there lies this contrast; on the one hand, the dimensionless quantum uncertainty of the potential box approaches classical dispersion only in the limit of large quantum numbers ( n → ∞ )—consistent with the correspondence principle. On the other hand, similar evaluations for bouncing ball and harmonic oscillator potentials are equal to their classical counterparts independent of n . This equality may hide the quantum feature of low energy levels. In the current study, we change the potential intervals in order to make them symmetric for the linear potential and non-symmetric for the quadratic potential. As a result, it is shown in this paper that the dimensionless quantum uncertainty of these potentials in the new potential intervals is expressed in terms of quantum number n . In other words, the uncertainty requires the correspondence principle in order to approach the classical limit. Therefore, it can be concluded that the dimensionless analysis, as a useful pedagogical method, does not take away the quantum feature of the n -dependence of quantum uncertainty in general. Moreover, our numerical calculations include the higher powers of the position for the potentials.
This chapter explores several cases of one-dimensional quantum motion, emphasizing the physical insights and analytical results. The particle in a rectangular potential well is examined, including the derivation of its quantized energy spectrum and normalization constant, particularly for the case of an infinitely deep well. Comparisons are drawn between the quantum and classical descriptions of motion. The analysis extends to a rectangular potential barrier, where reflection and transmission coefficients are calculated, and the phenomenon of quantum tunneling is explored. The tunnel effect paradox is addressed, along with its applications to α-decay and the cold emission of electrons from metals. Finally, the linear harmonic oscillator is treated in detail, including its energy eigenvalues, eigenstates, parity properties, and zero-point energy — the latter discussed as a consequence of the uncertainty principle and the wave-like behavior of microscopic particles. Metadata/Keywords: one-dimensional motion, rectangular potential well, α-decay, cold emission, quantum tunneling, linear harmonic oscillator, zero-point energy, parity, uncertainty principle, energy eigenstates
In (1), which deals with entropic uncertainty relationships, it is pointed out that x-variances of quantum bound states may possibly behave in a counterintuitive manner. In particular an example is given of two small boxes attached to the ends of a long pipe centered at zero so the average x is 0. If a particle has a probability of .5 to be in one box or the other a certain variance is found. If the particle can also be anywhere along the pipe or in the boxes the authors of (1) argue there is more uncertainty yet the variance decreases. Mathematically this occurs because there is now probability associated with small x values. We argue this is only counterintuitive if one considers probability distributions which decrease monotonically such as a Gaussian. In such a case, the variance increases if the particle is spread over a larger range. Such, however, does not seem to be the case if one has increased resolution which seems to be the case for quantum bound states such as a particle in a box with infinite potential walls or a quantum oscillator. Increased resolution occurs with increasing energy, but by the Heisenberg uncertainty principle the variance of p and x (or at least their product) increases. Increased resolution which seems to be linked to more order (less uncertainty) as argued in (1) can lead to an increased variance if probability only exists at distances far from the origin (i.e. high x) as in the closed two boxes example. As a result, the idea of uncertainty in the expression Heisenberg’s uncertainty principle seems to be based on the idea of how far the particle may be from the average value. As noted in (1) this means a large variance, but could represent high resolution i.e. a lack of uncertainty in finding the particle. We suggest there are two different kinds of uncertainty, Heisenberg’s uncertainty linked to the average and a resolution uncertainty. We examine the cases of a particle in a box and and the quantum oscillator.