Schrodinger Equation and Hamilton's Equations

Schrodinger Equation and Hamilton’s Equations Francesco R. Ruggeri Hanwell, N.B. Dec. 11, 2025 In (1), it is suggested that one may obtain the Schrodinger equation by using Hamilton’s equations as a starting point, i.e dp/dt = - dH/dq and dq/dt = dH/dp. (1) defines new variables phi(x,t) = ap+ibq and phi*(x,t) and writes: iC dphi/dt partial = {phi, H} and iC dphi*/dt partial ={ phi*, H}, where {} are the Poisson brackets. (1) then changes phi(x,t) into a stochastic variable which still follows these Hamilton-like relations iC dphi/dt = Integral dx H1(x1,x) phi(x) + b(x,t) where H= Integral dx1dx H1 phi*(x1)phi(x) and b(x,t is a random function. In other words, Hamilton’s theory with an added stochastic b(x,t) is applied to a stochastic variable in order to obtain the Schrodinger equation. The catch, however, is that (1) introduces a number of conditions and assumptions to simplify the problem and actually obtains iC dphi/dt = Integral dx H1(x1,x)phi(x) with b(x,t) gone. We suggest that this implies that one may apply the Hamilton-like form directly to a stochastic variable without b(x,t) present. To strengthen this argument, one may consider a bound state in which b(x,t) may be said to average to 0. Basically, the idea of (1) seems to be that quantum mechanics follows from introducing stochastic physics linked to usual Newtonian trajectory physics, in this case Hamilton’s theory. We argue that this is very similar to the classical action approach we used in (2) to obtain exp(-iEt+ipx) from the classical free particle action applied to stochastic motion and so we compare the two approaches. Approach of (2) to Finding Free Particle Quantum Mechanics We noted in (2) that one may propose a probability for free particles such that that the real value of the probability is 1 (all free particles have the same weight), but there exists an exp(i C1 E) and exp(i C2 p) (C1,C2 constant for units) such that one conserves momentum and at the same time ensures that any energy pair (Ei, Ej) or momentum pair (one dimension) pi,pj have the same probability. This is based on the idea that: exp(iC1 E1)exp(iC1 E2) = exp(iC1 E3)exp(iC1 E4) if E1+E2 = E3+E4 ((1)) and a similar result for p. This conservation holds in one frame. In the case of p, one may have p=m1v1=m2v2 (or the relativistic analogue). In a given frame, both p=m1v1 and p=m2v2 satisfy a conservation of momentum equation. Such is not the case in a Lorentz boosted frame as: P (with E1) —> g(v)p + g(v)vE1 and p (with E2) → g(v)p + g(v)vE2 with g(v)=1/sqrt(1-vv/cc) ((2)) As a result, we proposed (2): exp(-iEt+ipx) (one dimension) ((3)) -Et+px = classical Action in both the relativistic and nonrelativistic cases if x/t=v I.e. Lt = -mo t sqrt(1-vv/cc) = -Et+px ((4)) A= -Et+px, however, describes free particle motion. It is not supposed to be linked with stochastic behaviour, but it seems that it is by the above arguments and also by the fact that: A remains constant for x= x1 + hbar/p and t=t1+hbar/E ((5)) If x1,t1 are on the free particle trajectory x=vt, then x,t in ((5)) are not and represent stochastic fluctuations of the type needed to conserve momentum and energy. We note that in the above case, a Lorentz invariant involving linear E,p,x,t is required for both the momentum-energy conserving probability and for the action theory which must hold in any Lorentz boosted frame and so the same -Et+px appears for both. As a result, free particle quantum mechanics exp(-iEt+ipx) seems to arise from the classical action theory A = -Et+px (although one may independently derive it from the momentum-energy conserving probability and find that one uses -Et+px which happens to be the classical action.) We suggest it is this same notion of classical physics being applied to stochastic situations which is used in (1). We note that from exp(-iEt+ipx), one may obtain the Schrodinger equation through: i d/dt exp(-iEt) Sum over p a(p) exp(ipx) = Sum over p a(p) pp/2m exp(ipx) + V(x) Sum over p a(p)exp(ipx) ((5)) Hamilton Approach of (2) It seems that if one wishes to give a one sentence summary of the approach of (2), one might state that one may apply a Hamilton-like equation to a stochastic variable and obtain the Schrodinger equation. A priori, Hamilton’s equations describe deterministic physics (Newtonian mechanics) and so one might immediately object to them being applied to stochastic behaviour, but in the above section, we have argued that classical Action =Lt = -Et+px (v=x/t) for a free particle applies to stochastic free particle quantum mechanics. As a result, we examine the procedure of (2). We wish, however, to simplify it because we think that the main idea appears at the beginning and that one may obtain the result without most of the calculations presented in (1). We start with Hamilton’s equations: dp/dt = - dH/dq and dq/dt = dH/dp ((6)) In (1), new variables: phi= ap+ibx and phi* = ap-ibx are introduced and it is shown that: dphi/dt = Ci { phi , H } and dphi*/dt = iC { phi*, H} where {} are the Poisson brackets ((7)) C is a constant. ((7)) then seems to be generalized to a phi(x,t) and H to H= Integral dx dx1 H1(x,x1) phi*(x1) phi(x) ((8)) The form ((8)) is convenient for taking the derivative dH/dphi or dH/dphi*. The key equation then introduced in (1) is: iC dphi/dt = Integral dx H1(x1,x) phi(x) ((9)) In (1), phi(x,t) is then becomes a stochastic variable if ((9)) is replaced with: iC dphi/dt = Integral dx H1(x1,x) phi(x) + b(x,t) ((10)) where b(x,t) is a random number function with 0 mean A number of assumptions and calculations are then presented in (1) to argue that if phi(x,t) as a random variable preserves its norm Integral phi(x)phi*(x) and phi is a complex variable with constant modulus (i.e. only the phase is stochastic), then one arrives back at ((8)), which (1) calls the Schrodinger equation. We suggest that since one arrives back at ((8)), one should examine ((8)) in more detail with phi(x) as a stochastic variable and no b(x,t) random number generating function needed. To strengthen this assumption, one could consider a bound state problem and average over b(x,t) which then becomes 0. Thus, we suggest that one may consider ((9)) directly with phi(x,t) as a stochastic variable. We note that ((9)) is of the form of a Hamilton equation (to a certain degree). Thus, one tries to use a classical form and apply it to a stochastic variable. A priori, there is no reason to do this, except for the averaging of b(x,t) to 0 in the bound case, so (1) adds b(x,t) to make the equation stochastic, but then introduces various assumptions to arrive back at ((9)) with no b(x,t) present. This is the same as considering ((9)) as being applied to a stochastic variable from the beginning. In particular, there is no a priori reason given in (1) for arguing that phi(x,t) is a stochastic variable with a constant modulus and a stochastic phase. As a result, we consider the consequences of the Hamilton-like equation ((9)) applied to a stochastic function phi(x,t). What seems to occur in ((9)) is that there is an Et dependence in phi(x,t) which is linked with H1(x1,x)phi(x). The Hamiltonian brings out pp/2m (nonrelativistically) and so one may think of it as an operator. Only x is involved on the RHS of ((9)) so we assume that p is linked with x. If one wishes to have a Lorentz invariant argument, then it seems that it must be: -Et+px ((11)) Thus, ((9)) may be applied to a stochastic variable if one uses -Et+px as an argument. There is already a complex i present in ((9)) so one may use complex numbers. The question then becomes: Should phi(x,t) have a constant modulus? This is certainly a way in which to obtain a very simple solution to ((9)) namely: Phi = exp(-iEt+ipx) ((12)) Alternatively, one may take (1)’s equation ((10)) with the random number generating b(x,t) and argue that for a bound system, b(x,t) =0 on average and so one is really considering ((9)) on average. One would then have an Et argument linked with phi(x,t) on the LHS of ((9)), but H1 no longer brings out a pp/2m, but an average of pp/2m’s and involves V(x) as well. One would then have: phi(x,t) = exp(-iEt) Sum over p a(p)exp(ipx) and H= -1/2m d/dx d/dx + V(x) ((13)) The free particle idea of ((12)) is consistent with the more complicated bound state scenario ((13)). Conclusion In conclusion, in (2) we tried to introduce a free particle probability which has a real weight of 1 (all free particles have the same distribution value), but describes conservation of energy and momentum. We argued that one must respect special relativity and arrived at exp(-iEt+ipx). We then noted that A = Lagrangian * t= -Et+px (for v=x/t) is the relativistic and nonrelativistic classical action. In principle one could have started with classical action theory and noticed that t=t1+hbar/E and x=x1+hbar/p yield the same A value, suggesting fluctuations about the classical trajectory x=vt. Ultimately, this would also lead to exp(-iEt+ipx). The reason that -Et+ipx appears in both approaches is that it is the only Lorentz invariant linear in E,p,x,t . Thus, it seems that one may obtain quantum mechanics by using a classical equation A=-Et+px. We argue that (1) seems to apply this same idea of a classical equation to a stochastic variable, but in this case, Hamilton’s equations are used, i.e. dp/dt = -dH/dq and dq/dt = dH/dp. In (1), p→ phi=ap+ibq and q→ phi* = ap=ibq. This leads to: iC dphi/dt = { phi, H } and iC dphi*/dt = {phi*, H}, where {} are the Poisson brackets and C is a constant. The authors of (1), then consider phi(x,t) to be a function of x and t and write: iC dphi/dt = Integral dx H1(x1,x) phi(x,t) ((A)) where H = Integral dx dx1 H(x1,x) phi*(x1)phi(x). This form is convenient for dH/dphi. To make ((A)) stochastic, (1) adds a random number function b(x,t), but then after a number of assumptions finds that ((A)) is really the equation that holds. In other words, phi(x,t) is already stochastic in the deterministic Hamilton-like equation ((A)). We suggest that based on the ideas of the first section (first paragraph of the conclusion), one may consider this a priori. One may then find phi is a function of Et. For the RHS to depend solely on x, one may consider the variable px and H= -1/2m d/dx d/dx + V(x). Making the argument Lorentz invariant yields -Et+px as an argument and exp(-iEt+ipx) as a simple solution to ((A)) for the free particle case (V(x)=0). This may then be generalized to the V(x) bound state situation as we show. The overall idea, however, is to have a classical deterministic type equation (A=Lagrangian*t = -Et+px) or ((A)) apply to a stochastic function which seems to be associated with the deterministic trajectory, we argue. References 1. Oliviera, M. Derivation of the Schrodinger Equation from classical Stochastic Dynamics (2023) https://arxiv.org/pdf/2307.06461 2. Ruggeri, Francesco R. Quantum exp(-iEt+ipx) and Classical Action (preprint, zenodo, 2025)