The total energy of a particle (e.g., an electron in an atom, a free electron, an electron in an electric field, a conduction electron in a solid) is the sum of the V m p E Vpv VmvE potential and kinetic energies. Expressed in terms of momentum p, and mass m, this is stated: mvp since hfE h k f Considering movement in one dimension only, let t iE x ip i2 = 1 Now, the potential energy, V, may depend upon the position and time, and so in general, V = V(x,t).
The total energy of a particle (e.g., an electron in an atom, a free electron, an electron in an electric field, a conduction electron in a solid) is the sum of the V m p E Vpv VmvE potential and kinetic energies. Expressed in terms of momentum p, and mass m, this is stated: mvp since hfE h k f Considering movement in one dimension only, let t iE x ip i2 = 1 Now, the potential energy, V, may depend upon the position and time, and so in general, V = V(x,t).
The total energy of a particle (e.g., an electron in an atom, a free electron, an electron in an electric field, a conduction electron in a solid) is the sum of the 1 2 potential and kinetic energies. Expressed in terms of momentum p, and mass m, this is stated: mvp since Considering a one-dimensional case, Energy equation let t iE x ip i2 = 1 Now, the potential energy, V, may depend upon the position and time, and so in general, V = V(x,t).
Inadequate descriptions of atoms and molecules by the methods of classical physics led researchers to propose new ways to describe physical reality, giving birth to a totally new science, quantum mechanics. The methods of quantum mechanics are based on the introduction of a wave function, whose physical meaning is related to the probability of finding a certain particle, at a certain time in a volume element, positioned between x and x + dx in the x = direction, between y and y + dy in the y = direction, and between z and z + dz in the z = direction at certain time t. This wave function Ψ satisfies the Schrödinger equation, (−2mℏ2∇2+v)Ψ=EΨ,ℏ=2πh, or for short, HΨ = EΨ, where H, the Hamiltonian operator, is defined by the expression H=−2mℏ2∇2+V; h is Planck’s constant; ∇2 is the sum of the partial second derivatives with respect to x, y, and z; m is the mass of the particle; and V is the potential energy of the system. The Hamiltonian H represents the quantum equivalent of the sum of the kinetic energy and potential energy, with V being the potential energy operator and 2m−ℏ2∇2 the kinetic energy operator. Finally, E is the total energy of the system and is a number, not an operator.
Newton’s second law dp/dt = F(x)=-dV/dx may be written in terms of x alone i.e. dp/dx v(x) = -dV/dx or 1/(2m) d/dx pp = -dV/dx leading to the idea of kinetic energy pp/2m and potential energy V(x). In (1) we argued that there is velocity, rest mass mo, and momentum in quantum mechanics. A free particle, however, does not undergo smooth acceleration as in classical mechanics, and we argued there exists a momentum space wave exp(ipx) accompanying the moving particle. In such a scenario, force is considered to be probabilistic momentum hits of the form Vk exp(ikx) leading to a free particle momentum distribution W(x)= Sum over p a(p) exp(ipx). Both of these expressions involve interference due to momentum waves which do not track a particle in time. A mass flux mov or energy flux p=Ev relativistically seems to be the main consideration. Nevertheless, summing Vkexp(ikx) over k to obtain V(x) yields a function varying in x alone. This is a kind of average based on interference and Sum k Vk exp(ikx) is an average linked to momentum hits. We speculate that average momentum hits i.e. -d/dx V(x) are linked to momentum flux changes. In other words, even though an impulse is based on p and does not distinguish v i.e. p=m1v1=m2v2, momentum flux is velocity dependent and appears for an average momentum hit i.e. -dV/dx. In such a case (pave+dpave)(vave + dv ave) = 2 pave dv(ave) . This expression, however, contains both forward and backward motion which is part of quantum mechanics. Moving to an average picture seems to introduce time and direction i.e one uses pave d v(ave). This is already a change from a quantum to a classical scenario albeit for averages (which have not been specifically defined). Thus we consider pave dv(ave). If this is the case, then: d/dx pave 1/m space = -dV/dx or 1/(2m) pave pave = -d/dx V(x). Instead of treating pave pave as two separate averages one considers pave=prms or pavepave=<pp>. Thus the quantum bound picture (at any energy level) becomes a classical type of scenario which includes time and velocity. Note that for a two slit interference pattern, one does not need velocity and time because the interaction with the potentials from both slits is based on exp(ipx). One does not even need the form of the potential. Interference is the same for p=m1v1=m2v2 (even though the process takes place during different time intervals, the pattern is not affected). In other words, we argue that Newton’s second law might follow from a picture of momentum hits in which a flux i.e. p=mov is treated as a quantity which in an average scenario leads to a second flux i.e. an average momentum flux which reintroduces direction, time and velocity.. This is somewhat like a hydrodynamics scenario.
Abstract: The well-known variation method of calculating the total energy of a two-electron atom is generalized to the case of any set of quantum numbers of electrons in spaces of three or fewer dimensions, yielding the corresponding expressions for energy and screening constant. From these results, the particular cases of the basic and the lowest excited states follow, as earlier considered by us and other authors. We clarify the correct use of these results. Moreover, we generalize the standard relations connecting the average values of kinetic and potential energy with the total energy of an electron in a nucleus field in three dimension for any quantum state and extend it to subspaces of one and two dimensions in a three-dimensional space. The results of this work could be a useful addition to a university course on quantum mechanics.
In classical statistical mechanics, C exp[ -( mv*v/2 + V(x))/T) represents density i.e. the probability for a particle to have a velocity v and be at x. If applies this probability to Shannon’s entropy formula, one obtains C/T ( mv*v/2 + V(x)) exp[ -( mv*v/2 + V(x))/T) + exp[ -( mv*v/2 + V(x))/T) ln(C). Thus, it seems energy times density + density times ln(C) is the entropy density, a well known result in classical statistical mechanics, although sometimes not explicitly stated. The first term integrated over x is essentially the kinetic energy, while the second integrated over velocity yields the potential energy term. Thus, the first term partial density seems to be a purely momentum dependent kinetic energy piece times probability and the second a purely x dependent potential energy piece. (There is still an integral over velocity for the first term and one over x for the second to obtain the full entropy.) In quantum mechanics, we argue that a(p1)exp(ip1 x) a(p2) exp(ip2 x) serves as the probability where W(x)=Sum over p a(p) exp(ipx) = wavefunction. This leads to an expression for entropy density with two terms, one of which may be immediately integrated over momentum and the other over space. If one wishes to make a link to the classical statistical mechanical case, one may ask what requirement is necessary for the momentum related piece to represent average kinetic energy and the spatial part, average potential energy. The requirement, we find, yields the wavefunction of the ground state harmonic oscillator. For other cases, the partial density is not equivalent to average energy. Thus, we try to establish a link between entropy density and energy density, but require the kinetic energy related term to be strictly a function of momentum and the potential related term strictly one of x. In quantum mechanics, one may form averages at a point x. For example, KEave(x)= [Sum over p p*p/2m a(p) exp(ipx)]/W(x). If one examines a p/-p pair with a(p)=a(-p) (for example), cos(px) terms arise which are positive at some x and negative at others. We argued in previous notes that this represents stochastic flow in the opposite direction from that expected by -dV/dx. In other words, -dV/dx is an average which consists of stochastic force addends which may give momentum hits to the right or left, but which add to yield -dV/dx. As a result, one may have negative contributions to the kinetic energy (and density as well). In this note, we argue one may also have negative contributions to the entropy density due to this phenomenon.
The notion of temporal and spatial interactions exists in Newtonian mechanics and we suggest it is useful to compare the two as they differ even though time and x are both considered co-ordinates of spacetime. In particular, space has two directions for every spatial co-ordinate (positive and negative) and so one may move backwards in space. This is associated with a force which may be positive or negative. Even though one may write a spatial equation in terms of energy, if kinetic energy becomes zero due to a negative force, then after this, the particle moves in the negative direction. As a result, one has a potential or interaction energy V(x) which is linked to force = -grad V(r). (We only consider V(x), not V(x,t) here). Things are different with interactions which take place in time only, even though these too may be considered in terms of energy. In general, these seem to be more statistical in nature. In particular, a bound system may absorb a photon. This is an interaction in time, but all subsequent interactions are in increasing time. There is no going backwards in time. An excited state may lose energy or gain more energy. There is an interaction, but no force backwards in time. In such a case, the energy of the photons is their state energy , while V(x) is not a state energy of the system creating it. For example, a charged particle has a V(r) = -kq/r, but its state energy is .5 eo Electric field dot Electric field, where Electric field = -grad V(r). One may already see differences in time interactions in classical statistical mechanics even if a V(x) exists. As a result, of the differences between positive and negative motion in x and only positive motion in t, one may have a localized particle in space and this leads to boundary conditions which yield discrete energy levels. At the same time, a distribution in free particles p’s in the wavefunction W(x) = Sum over p a(p)exp(ipx) leads to a spatial distribution W*(x)W(x). At the same time, interactions in time (with photons for example), lead to a distribution in atom states in the wavefunction, W(x,t) = Sum over n exp(-En/T) Wn(x) and a time distribution Integral dx W*(x,t)W(x,t).