In (1), an approximate method of obtaining the ground state energy is presented by mapping a Schrodinger equation with a given potential V(x) into one with delta function potential with an appropriate weight. Various examples are presented in both (1) and (2). It should be noted that although this approach yields the exact ground state energy for the oscillator it does not yield a correct answer for a particle in a box with infinite potentials at the wall. In this note, we try to investigate a possible physical argument for a mapping into a delta function and argue it is linked to the Heisenberg uncertainty principle. We also suggest that the weighting used in (1) i.e. U1= Integral (x1,x2) V(x)dx where (x1,x2) are classical turning points, should be multiplied by a constant b which ultimately cancels in the expression for the ground state energy.
In (1), an approximate method of obtaining the ground state energy is presented by mapping a Schrodinger equation with a given potential V(x) into one with delta function potential with an appropriate weight. Various examples are presented in both (1) and (2). It should be noted that although this approach yields the exact ground state energy for the oscillator it does not yield a correct answer for a particle in a box with infinite potentials at the wall. In this note, we try to investigate a possible physical argument for a mapping into a delta function and argue it is linked to the Heisenberg uncertainty principle. We also suggest that the weighting used in (1) i.e. U1= Integral (x1,x2) V(x)dx where (x1,x2) are classical turning points, should be multiplied by a constant b which ultimately cancels in the expression for the ground state energy.
This article establishes the proof of the Schrödinger equation for numerous quantum systems, utilizing Heisenberg's uncertainty principle. The Fourier transform connects functions in the time and frequency domains, resulting in the mathematical inequality that is the foundation of the uncertainty principle. In the part of Methods and Theory, the article derives the uncertainty principle through Fourier transforms by defining the mean and variance of angular frequency and time, and subsequently expanding the integral. This establishes the fundamental connection between time and frequency domains, illustrating the constraints imposed by quantum mechanics. In the part of Results and Application, the article applies the uncertainty principle to derive the Schrödinger equation under different conditions: free particle, particle in a box, harmonic oscillator, and hydrogen atom. For each case, the article assumes wave function solutions, uses the uncertainty in position and momentum to estimate kinetic and potential energies, and shows that the total energy matches the ground state energy derived from the Schrödinger equation. The results highlight the critical role of Heisenberg's uncertainty principle in understanding key aspects of quantum mechanics, providing a unified framework for these diverse systems.
Mathematically, it is possible to write a potential V(x) in classical mechanics as a Fourier series. It is not guaranteed, however, that this series have any physical meaning. In this note, we argue that it has meaning if one thinks of the potential as an ensemble of different momentum sources, each with a different weight. Then, placing a particle in such a potential could yield an ensemble of momenta f(p). This ensemble W(x) at this point is not associated with a Fourier series, but it is an equilibrium ensemble and can also be thought of as a potential, it seems, which has different weights associated with different momenta. If this is the case, this ensemble can also be written as a Fourier series W(x) = sum on p f(p) exp(ipx). Now, the ensemble W(x) created by the potential V(x) in the first place, can interact with the potential again and again. This can be described by writing V(x)W(x). This combination should return the original ensemble W(x) up to a constant if one has an equilibrium situation. Now V(x)=Energy-average Kinetic energy which leads to the Schrodinger equation.
Heisenberg’s uncertainty principle for position-momentum delta x delta p >= hbar/2 is considered as an important statement regarding quantum mechanics i.e. one cannot know both p and x exactly. Delta x and delta p stand out because in classical mechanics one has an eact p and x at each t. Mathematically, however, delta x and delta p are standard deviations which may also be calculated in classical statistical mechanics, for example with the Maxwell-Boltzmann distribution. In particular, if one considers the ground state of a simple harmonic oscillator, the quantum P(x) proportional to exp(-pp/a) and P(p) proportional to exp(-xxa) match the classical statistical probabilities P(x) and P(p) for a fixed value of T proportional to sqrt(k/m) where k is the spring constant. Thus for this special T, delta x delta p for a classical statistical mechanical gas is identical to delta x delta p for the ground state quantum oscillator. The probability is normalized the same way for both. The physics of the two situations, however, is completely different. Thus the Heisenberg uncertainty principle does not necessarily describe quantum mechanical behaviour alone. If one uses entropic uncertainty, i.e. Sp + Sx where S is Shannon’s entropy, once again one uses P(p)=a*(p)a(p) and P(x)=W*(x)W(x) where W(x)=wavefunction = Sum over p a(p)exp(ipx). In principle, one could also calculate such an entropy for a classical gas i.e. use P(x)P(p) in Shannon’s entropy. The result Sp+Sx does not imply one does not know about p if one knows x i.e. the usual quantum statement.
In a previous note (1), we argued that a potential V(x) may be written as Sum over k V(k) exp(ikx). Thus, -dV/dx is only an average force which may in fact be zero, yet -d/dx V(k) exp(ikx) is not zero leading to possible forward and backward motion/acceleration (due to each V(k) exp(ikx) which act at a different times). This gives rise to a momentum distribution P(p/x) =a(p) exp(ikx)/W(x) (W(x)=wavefunction) at each x. In this note, we wish to link these ideas to the Bohm Aharonov effect by considering two recent (2018, 2020) examples from the literature. In the first (2), the magnetic potential A(r) is written in a Fourier series in the same manner as V(k)exp(ikx). Thus, although grad A = B (magnetic field) = 0, grad A(k) exp(ikx) should not be zero and so there may be hits from “virtual photons”. (2) applies perturbation theory to find a correction to energy E and then considers exp(i Integral over t delta E dt) as in adiabatic theory. The second approach (3) applies the usual Hamiltonian containing the magnetic potential A, but imagines a little box at various xi positions such that A(xi) is a constant at each xi. In such a case, one may show directly that W(x)= W(x=0) exp( ie/hbar Integral xo to x1 A(x) dx) which yields the Ehrenberg–Siday–Aharonov–Bohm effect. In this approach, one essentially has a different constant potential at each point and hence a different energy. In a previous note, we argued that even though average kinetic energy may remain the same, a change in constant potential using Sum over k V(k) exp(ikx) gives rise to different physical effects which try to argue are related to the Bohm Aharonov effect. We also try to consider a more general case.
In bound state quantum mechanics, one has statistical observables (based on P(x) where P is probability) such as spatial density d(x), potential energy density V(x)d(x) and kinetic energy density .5mv(x)v(x) d(x), where v(x) is the classical velocity obtained from .5mv(x)v(x) + V(x) = E. (Kinetic energy density can also be written as -1/2m W(x) d/dx d/dx W(x) where W(x) is the wavefunction). The energy of a state can be obtained by integrating these statistical observables i.e. Integral dx d(x) V(x) + Integral dx d(x) .5mv(x)v(x) = E. If one performs this integration for two different states, one may calculate the energy difference. This begs the question: why does quantum mechanics deal with conditional probabilities such as P(p/x) directly related to W(x)? The issue perhaps begins with the appearance of V(x) and E in the time independent Schrodinger equation. Kinetic energy (not kinetic energy density) should appear and be somehow linked to some motion. A question arises as to why the energy density is proportional to the spatial density i.e. Energy density = E d(x)? This is directly linked to the time-independent Schrodinger equation, but one may try to argue that classical energy conservation .5mv(x)v(x) + V(x) = E at each point x should hold in quantum mechanics with .5mv(x)v(x) equal to Sum over p p*p/2m P(p/x) where P(p/x) = fp exp(ipx)/W(x). If one, however, considers excited states written as Wn(x)= Wo(x)Hn(x) where Wo(x) is the ground state wavefunction, one finds that: -1/2m d/dx d/dx Hn(x) - 1/m [d/dx Wo(x)]/Wo(x) [d/dx Hn(x)] = E1 Hn(x) ((1)) In other words, an eigenvalue equation appears again with no V(x) present. If one assumes Hn(x) has physical meaning (i.e. is related to a phonon or photon), then ((1)) may suggest there is a physical principle in place, i.e. the idea that one must have a “constant” energy at each point x. Here E1 is strictly kinetic energy (although not all of it). This “principle” has consequences as it restricts the values of E1 (i.e. discrete eigenvalues) and leads often to a periodic Hn(x) which has direct effect on kinetic energy density and potential energy density. Thus, having Hn(x) interact with Wo(x) to produce a constant energy E1 times Hn(x) does affect kinetic and energy densities at each x point, just not kinetic energy itself (it is constant i.e. E1). The purpose of this note is to examine this possible principle behind quantum mechanics.
The usual Heisenberg uncertainty relation, ΔX ΔP ≥ ℏ/2, may be replaced by an exact equality for suitably chosen measures of position and momentum uncertainty, which is valid for all wavefunctions. This exactuncertainty relation, δX ΔPnc ≡ ℏ/2, can be generalised to other pairs of conjugate observables such as photon number and phase, and is sufficiently strong to provide the basis for moving from classical mechanics to quantum mechanics. In particular, the assumption of a nonclassical momentum fluctuation, having a strength which scales inversely with uncertainty in position, leads from the classical equations of motion to the Schrödinger equation.