Based on the assumption that the probability density of finding a free particle is independent of position, we infer the form of the eigenfunction for the free particle, ⟨x|p⟩=exp(ipx/ℏ)/2πℏ. The canonical commutation relation between the momentum and position operators and the Ehrenfest theorem in the free particle case are derived solely from differentiation of the delta function and the form of ⟨x|p⟩.
Based on the assumption that the probability density of finding a free particle is independent of position, we infer the form of the eigenfunction for the free particle, ⟨x|p⟩=exp(ipx/ℏ)/2πℏ. The canonical commutation relation between the momentum and position operators and the Ehrenfest theorem in the free particle case are derived solely from differentiation of the delta function and the form of ⟨x|p⟩.
The canonical commutation relation between the quantum mechanical operators of momentum and coordinate and the Ehrenfest theorem in the free particle case are derived solely from the rule of differentiation of the delta function [xδ′(x)=−δ(x)].
This chapter introduces an elementary form of the fundamental axioms of quantum mechanics, focusing on the structure and interpretation of state functions within a vector space framework. The properties of linear operators acting on state functions are explored, along with the physical meaning of the state function itself. From this interpretation, boundary conditions and normalization requirements are derived. The chapter presents key results such as the normalization theorem, the time derivative of a physical observable, and Ehrenfest's theorem. It also defines probability density and probability current density, deriving the continuity equation for probability. The uncertainty principle is formally established, and energy conservation is addressed. The eigenvalues and eigenstates of the momentum operator are determined, and normalization using the Dirac delta function is introduced. The free particle wavefunction is derived, concluding with a discussion on the classical limit of quantum mechanics. Metadata/Keywords: quantum axioms, state function, linear operators, probability density, probability current, Ehrenfest's theorem, uncertainty principle, Dirac delta function, momentum eigenstates, classical limit
In previous notes, we introduced a complex probability into Newtonian two-body elastic scattering to ensure that for a given (e1,e2) and (p1, p2) (momentum vectors), any (ei, ej) (pi, pj) have the same probability to occur. This ultimately led to the Lorentz invariant probability exp(-iEt+i p dot r). Here we begin with the delta functions delta( pinitial - pfinal) and delta( Einital - Final) and see that one may write these as Integral da C exp(-i pfinal + i p initial) and a similar expression for E. Here a is simply a math variable. (Note: This is only one math form for a delta function, but this particular form will prove useful from physical considerations later.) The delta function may be used in an expression Integral delta(pinitial - pfinal) dpfinal and only pfinal values which satisfy conservation of momentum yield a nonzero integral. Furthermore, this integral may be multiplied by various functions and still maintain conservation of momentum. At this point, the above is simply a calculation formalism. The question we ask is whether one may make the argument that a particle itself is responsible for conservation of momentum in an interaction. We suggest this may be the case based on Newton’s notion of action-reaction. It is not simply that one applies a force to a particle and it changes. The particle itself resists and this is the idea of inertia as well as action-reaction. The particle itself is strongly involved in a reaction and we suggest that a particle carries a property of ensuring conservation of momentum and energy itself. It is not something that is imposed by the interaction, otherwise there would be no need for the notion of action-reaction. The particle reacts and the manner in which it reacts leads to conservation of momentum and energy. We suggest that there seem to be grounds for considering a particle to actually be the driver of energy and momentum conservation and that this actually extends to further features, namely probabilistic results in interactions which do not appear in Newtonian mechanics. This property might then be taken as exp(i p a) and exp(i E b), where a and b are math variables used to establish the delta function. One, however, cannot really establish a property of a particle if math running variables a and b are present. The entire expression should consist of particle state variables. This seems to suggest the consideration of Lorentz invariance because E and p change when the particle is seen from a frame moving with -v. One may create a state based exp(-iEt+ipx) energy-momentum conserving property of a particle which is Lorentz invariant. We next note that p and E are linked to interactions, but ones that differ. In particular, p is linked with an impulse which has nothing to do with velocity , i.e. E. On the other hand, E = .5movv in the nonrelativistic limit and is dependent on v, i.e. a m1v1= m2v2 have the same impulse, but different kinetic energies. Thus, we suggest that exp(-iEt) and exp(ipx) be used separately depending on the type of interaction one considers. In a quantum single particle bound state, one is interested in impulse hits with the potential we argue and so does not need to consider exp(-iEt) as one cannot measure the particle moving through the potential in a classical manner. We argue that a constraint (momentum and energy conservation) may actually be due to properties of the particle and that this ultimately follows from Newton’s action-reaction. In (1), we argued that exp(-iEt+ipx) follows from Lorentz invariance which is based on the Newtonian idea of work increasing energy, i.e. 1/sqrt(1-vv/cc) and A= -Et+px have minus signs for this very Newtonian reason. We argued that dx = hbar/p and dt=hbar/E follow as fluctuations which leave A unchanged due to the presence of the minus sign.Here we again argue that a Newtonian idea is responsible for exp(-iEt+ipx), namely action-reaction. We show that the two approaches are identical.
We have argued in previous notes that both Newtonian mechanics and free particle quantum mechanics arise from special relativity. Thus, special relativity gives an expression for free particle energy and momentum and these are related to concepts of work. The Lorentz invariant -Et+px shows that x,t and x+hbar/p, t+hbar/E give the same solution to this potential-like term showing that there is stochasticity in the position and time delivery of interaction hits. In other words, even though the center-of-mass moves as x=vt, a momentum hit (proportional to rest mass mo) does not necessarily act at the center-of-mass. This gives the impression of shifts in probability, kinetic energy etc in the case of interference, i.e. interactions. For a single free particle, there is no interaction and -1/2m d/dx d/dx exp(ipx) = pp/2m exp(ipx). We argue that exp(ipx) is a probability and that in general one may add independent probabilities in an OR situation as in a 2-slit interference case. A bound state, however, is different because even though one adds weighted exp(ipx)s, they are part of a resonance or equilibrium and this is no longer an independent OR situation. We suggest that one uses a conditional probability when computing a Newtonian nonrelativistic conservation of energy equation to try to remove as much of the quantum features from this equation. As a result, one has: KEclassical(x) + V(x) = E = Sum over p pp/2m P(p/x) + V(x). One cannot bypass the fact that a quantum particle interacts differently from a classical one and that this involves a probabilistic granularity (wavelength), but classical notions of energy and energy conservation still exist. As a result, the bound state wavefunction Wn(x) = Sum over p an(p) exp(ipx) is not a usual OR sum, but is rather 1 over a normalization factor for an equilibrium picture. As a result, one cannot use Wn1(x)Wn2(x) = W(n1+n2) (x) as one would for independent probabilities. Furthermore, given that one has Wn(x) and Wn1(x)+Wn2(x) which are both of the form Sum over p a(p)exp(ipx), and both represent average energies, one needs a way to distinguish a resonance/equilibrium from a sum of resonance/equilibrium normalization factors. We suggest that the orthogonality of resonance Wn(x)’s provides this feature. The standard textbook explanation is based on Hermitian operators, but in the nonrelativistic case, we argue that one may obtain the result based on integration by parts based on the form of the p operator = -grad. Thus, it is ultimately the form exp(ipx) which leads to orthogonal bound state Wn(x)’s, together with the boundary condition that Wn(x)-->0 of x-> +infinite and -infinite.
In classical physics, one states in words the existence of conservation of momentum and energy. An example would be an elastic collision of two particles. In particular, this conservation is imposed, it does not mathematically appear unless one describes the process by say a Kronecker delta, i.e. delta( momentum vector before, momentum vector after) or delta(energy before, energy after). Even in such a case, the Kronecker delta is isolated and so one still has to manually ensure that energies and momenta chosen in a calculation ensure conservation. A Kronecker delta is mathematically equivalent to the dot product of two orthonormal states in linear algebra. This implies an operator which may obtain the particular value in question, momentum vector or energy, when operating on the state. A Hermitian matrix operator is guaranteed to have real eigenvalues and orthogonal eigenvectors, which is required. Thus, mathematically, one may postulate energy and momentum states and corresponding operators because this automatically creates the Kronecker delta which describes conservation in classical processes. This approach holds both nonrelativistically and relativistically. The question becomes: How does one specifically describe the operator and its state? For both a relativistic and nonrelativistic free particle classical action, which describes motion, one may write: A= -Et+px where E,p are the relativistic free particle values in one case and E=p dot p/2mo and p=mov in the other. One may note that E and p are values which may be conserved in interactions. The action shows that the operators -d/dt partial and d/dx partial yield E and p respectively, but one does not have an eigenstate. This may be obtained using id/dt partial exp(-iEt+ipx) = E exp(-iEt+ipx) and -id/dx partial exp(-iEt+ipx) = p exp(-iEt+ipx). Because one uses partial derivatives, t and x are treated as independent relative to the time and space translation operators d/dt partial and d/dx partial. As a result, we argue that by mathematically linking a Kronecker delta associated with classical conservation with linear algebra, i.e. momentum and energy states and finding a corresponding operator by examining the classical action leads to the quantum free particle wavefunction exp(-iEt+ipx). One may apply this to an elastic two particle collision to see that the Kronecker delta and its associated state/operator form describe conservation of both energy and momentum.One may ask whether exp(ip dot r) has any physical relevance. Two-slit interference experiments using particles with rest mass shows that it does.