An interpretation of the squared amplitude of a solution of the Schrodinger wave equation. Tim C Jenkins.

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1 An interpretation of the squared amplitude of a solution of the Schrodinger wave equation Tim C Jenkins tjenkins@bigpond.net.au Keywords: Schrodinger wave equation, stochastic probability density, Chi-Square distribution Abstract It is found that that from a mathematical standpoint the probability density given by a solution of the Schrodinger wave equation is determined by an underlying random variable which has a Chi-Square distribution with one degree of freedom, and that the resulting mean probability density agrees with its deterministic value in quantum mechanics. As a result quantum probability and interference patterns in particular, would obey the normal laws of probability. 1 Introduction In quantum mechanics the probability distribution of the location of a particle at time t is described by a wave function ( ), which is a function of the three spatial coordinates and time and which is defined and continuous everywhere. The squared amplitude of the wave is ( ) ( ) ( ) and the probability density ( ) associated with the location of the particle is given by ( ) ( ) ( ) ( ) where is the complex conjugate of and ( ) is a normalisation constant which varies with t. Provided that is finite C rescales the squared amplitude to a probability density by satisfying and reflects the fact that the particle exists somewhere at time t. 1 This paper considers, without loss of generality, a particle of mass 1 that is governed by a wave function that is a superposition of two solutions of a one-dimensional Schrodinger equation (with x being the spatial coordinate). It concludes that, at least from a mathematical standpoint, the probability density ( ) associated with a solution of the Schrodinger equation is not fixed at ( ) but rather is determined by an underlying random variable that 1 Where the context permits is used as shorthand for ( ), and likewise for other functions of. 1

2 has a Chi-Square distribution with one degree of freedom, and that the resulting mean probability density agrees with its deterministic value in quantum mechanics. 2 The function The one dimensional Schrodinger equation in our setup is as follows ( ) Consider two distinct solutions of this equation, expressed in polar form, namely, and where are the respective wave amplitudes and are non-negative real functions of space and time; are the respective squared amplitudes; and are the respective wave phases and are also real functions of space and time [1]. Now form a wave function solution that is the sum of these wave functions, namely and calculate the squared amplitude (using and ): ( )( ) [ ( ) ] (1) The furthest term on the right describes the interference between the waves. Its sign and size are variable and depend on the phase difference at each point in space and time [2]. The remarkable feature of (1) is that the cosine is mathematically equivalent to and has all the attributes of a coefficient of correlation [3]. It follows, from the standard result in statistics for the variance of the sum of two correlated random variables [4], that is equivalent to this variance, where the two variables have variance and respectively and a correlation coefficient of [ ( ) ]. 2 3 The significance of as a variance Quantum mechanics is a physical system that is described in probabilistic terms and it is therefore impossible to ignore the apparent coincidence that mathematically the variance of the sum of two correlated random variables. is 2 The same applies when there are more than two wave solutions superposed, in which case each cosine in the cross-product terms in the squared amplitude corresponds to an equivalent correlation. 2

3 However, the undoubted empirical success of quantum mechanics establishes that is also the squared amplitude of the superposed sum of the two wave functions and that this squared amplitude, after normalisation, gives the probability density ( ) associated with the particle being at the point ( ). This means that before normalisation ( ) ( ) is notionally the expected number of particles in the interval at time t, out of the total number of particles described by the wave function prior to normalisation, this number being equal to the normalisation constant ( ). In a similar fashion ( ) ( ) is particle density at the point ( ), again notionally because ultimately, once the solution is normalised, the wave will describe the probability distribution of only one particle. An interpretation that would reconcile the mathematical and physical standpoints would then be that before normalisation the squared amplitude (or notional particle density) is the mean of a random variable, and that this mean is the same as the variance of another closely related and relevant random variable. The only probability distribution that allows us to satisfy this requirement is the Chi-Square distribution with one degree of freedom ( ). From a standard result in statistics, if a random variable X is ( ) then the random variable is ( ) [5]. It follows immediately from this that is also ( ). The mean of ( ) so that the mean of is 1 and it follows that the mean of the random variable is, namely the variance of the random variable. This is what is required to reconcile the mathematical and physical standpoints, as follows. Assume that the squared amplitude before normalisation (which can be thought of as particle density) at each point is stochastic, not deterministic, and that it is a random variable which is the square of a another random variable where R is distributed ( ). It can be seen that is non-negative and the stochastic analogue of wave amplitude. From the previous paragraph it follow that is ( ) and the mean of is, which reconciles the two standpoints. By way of example, in the case of the superposed wave described earlier, the variance of the random variable equals the variance of the sum of the random variables, which from (1) and from the equivalence relation between the coefficient of correlation and cosine is [ ( ) ]. The distribution of is then ( ) and the mean squared amplitude is therefore [ ( ) ] which agrees with its deterministic value in quantum mechanics. 3

4 4 Stochastic probability density By extension of the forgoing analysis the following results apply to any number of superposed wave function solutions of the wave equation at each point ( ) in space and time we have so that as noted already. ( ) ( ) ( ) ( ) (2) ( ( ) ) ( ) ( ) (3) Because the squared amplitude variable is stochastic, it is realised at each point in space at time t and likewise the probability density derived from it, but of course the particle itself will be realised at only one point, when it is observed. The realised probability density ( ) at a spatial point ( ) at time t is therefore determined from the realised squared amplitude ( ) by dividing it by ( ), the realised normalisation constant, which sums all the realised squared amplitudes at time t across space. That is, ( ) ( ) ( ), hence and from (2) ( ) ( ) ( ) ( )[ ( ) ( ) ( )] ( )[ ( ) ( ) ( )] ( ) (4) To proceed it is reasonable to assume that squared amplitude (or notional particle density), and hence probability density, is realised in infinitesimal pieces in disjoint volumes of space and that ( ) is the probability density piece around the point ( ). From (4) the realisation of each piece is independent of the simultaneous realisation of every other piece and, because the impact of any realised squared amplitude piece has an infinitesimal influence on ( ) ( ( ) ( )) ( ( )) ( ( )) (5) Also ( ( )) ( ) (6) so that from (4), (5) and (6), and recalling that the mean of ( ) is 1, we have 4

5 ( ( )) ( ) ( ) ( ( )) ( ) ( ) ( ) (7) showing that the expected probability density agrees with the deterministic probability density in quantum mechanics. Finally, consider a set of realised probability densities. Begin by noting that if each of N random variables ( ) is distributed ( ) then ( ) is distributed ( ) [6]. Also the Chi-Square distribution with N degrees of freedom has mean and standard deviation so that, so that the ratio of the standard deviation to the mean is therefore ( ) ( ) ( ). (8) Let W be the ratio of the realised squared amplitude random variable to expected, that is, and consider N infinitesimal simultaneous realisations of W across space at time t. From (7) each realisation has an expected value of 1 and from the previous paragraph their sum ( ) is distributed ( ). It then follows from (8) that the average realisation ( ) will approach 1. This suggests that across the large number of infinitesimal pieces of space involved in this interpretation the average fluctuation from expected would closely approximate zero and that stochastic behaviour when viewed on a large enough scale would be consistent with what is expected using a deterministic interpretation of quantum mechanics. 5 The physical meaning of - a thought experiment The mathematical hypothesis described in this paper can be summarised as follows: 1. At each infinitesimal piece of space there is a stochastic equivalent of wave amplitude where the distribution of R is Normal with a mean of zero and a variance equal to the deterministic wave s squared amplitude. 2. The same is true everywhere else, with the random variable in one piece of space being independent of in another. 3. The ratio of the stochastic equivalent of squared amplitude to the deterministic wave s squared amplitude has a Chi-Square distribution with 1 degree of freedom. 4. The expected value of equals the deterministic wave s squared amplitude. In order to give this hypothesis some physical context it would be necessary as a first step to ascribe some relevant physical meaning to the random variable in an infinitesimal piece of space and then as a second step to establish that this meaning leads to the conclusion that when this random variable is squared and normalised the result is the probability density that the particle will be realised there. To emulate such a program, and to attempt to glimpse a physical meaning of the random variables and, we will conduct a thought experiment with a simple electromagnetic 5

6 wave with electric field amplitude E. Drawing on the basic physics of electromagnetic waves energy density u is given by where is the electric permittivity of free space, and the energy in a volume (which is the wavefront area times the length ) is given by ( ). The energy density at a point and the energy content of a volume of space are therefore proportional to with the same constants of proportionality at each point and each volume dv respectively. Now consider a setup in which two electromagnetic waves with the same polarisation are emitted from two sources. Suppose that they have amplitudes and respectively and a phase difference of when they arrive at some point. The superposition of the two fields at that point is a wave function with amplitude that depends on the individual amplitudes and the phase difference. As is well known, by using a phasor representation or otherwise, at the point is given by which, by putting ( ) in (1), can be seen to be in formal correspondence to the deterministic squared amplitude given by a solution of a Schrodinger wave equation. Although the latter does not apply to a photon, this formal correspondence means that a thought experiment can be conducted to consider what it would mean for an electromagnetic wave if the mathematical hypothesis did apply to it, thereby giving the latter some physical context to judge it by. As the first step assume, for the purposes of the thought experiment only, that the electric field amplitude at the point in question is not smooth and deterministic but is a random variable where R is Normally distributed with a mean of zero and a variance equal to. From previous sections is distributed ( ), the mean of is and the mean energy density is. Assume as the second step of the thought experiment that the probability of the photon being realised in the infinitesimal volume is proportional to the energy content there, which in turn is proportional to, and it follows that after normalisation would give that density. It also follows that the expected probability density would be proportional to which in turn is proportional to the expected energy density. Coming now to the point of the thought experiment, what would it mean for the behaviour of an electromagnetic wave of deterministic amplitude E if according to our hypothesis its amplitude is replaced by a random variable at each point, and R is ( )? From the moments of the half-normal distribution [7] ( ) ( ) = , and ( ) ( ) = The range from one standard deviation below to one above the mean is therefore E to E. The wave s amplitude would fluctuate considerably from point to point on an infinitesimal scale. 6

7 It is not being suggested that this happens to an electromagnetic wave, which is only being used as a tangible touchstone for the conduct of the thought experiment. On the other hand, it is being suggested that if probability in quantum mechanics is to obey the laws of probability then the amplitude of a wave that is a solution of the Schrodinger wave equation would fluctuate to this extent on an infinitesimal scale. Thus, if amplitude is realised in an infinitesimal piece of space and falls in the equivalent range to that discussed above, namely to , the squared amplitude would fall in the corresponding place in the range to We can therefore round off the experiment by saying that a tangible parallel to the random variable is that it is a stochastic quantum analogue of the amplitude of an electromagnetic wave, and is an analogue of that wave s energy density. 6 Conclusion The implication of this interpretation is that the normalised squared amplitude of a solution of the wave equation in quantum mechanics would not be thought of as fixing the probability density ( ) at a point ( ) in space and time but that instead the probability density would be thought of as a random variable the mean of which equals the normalised squared amplitude. Of course quantum mechanics works perfectly well without such a mathematical reconciliation. Nevertheless the interpretation in this paper may help open up new lines of theoretical enquiry by positing that the distribution of the ratio of the realised squared amplitude to its expected value is ( ). At the very least it shows that the interference patterns of a superposed wave equation solution in quantum mechanics may come from the variance of a linear combination of corresponding random variables and that it is these variances that exhibit interference (i.e. correlation), not probability densities. Mathematically, interference between probability densities exists nowhere in probability theory and is a construct that is unique to quantum mechanics. The interpretation of the squared amplitude in this paper, on the other hand, would allow quantum mechanics to obey the laws of probability. References [1] Holland, P. R. The Quantum Theory of Motion: an account of the de Broglie-Bohm causal interpretation of quantum mechanics, Paperback ed., Cambridge University Press, Cambridge, 1997, pp [2] Holland, P. R. The Quantum Theory of Motion: an account of the de Broglie-Bohm causal interpretation of quantum mechanics, Paperback ed., Cambridge University Press, Cambridge, 1997, pp

8 [3] Papoulis, A. Probability, Random Variables, and Stochastic Processes, 3 rd ed., McGraw-Hill Inc., Singapore, 1991, p 154 [4] Papoulis, A. Probability, Random Variables, and Stochastic Processes, 3 rd ed., McGraw-Hill Inc., Singapore, 1991, p 155 [5] Hogg, R. V and Tanis, E.A. Probability and Statistical Inference, 4 th ed., McMillan Publishing Company, New York, 1993, pp [6] Weisstein, E.W. Chi-Squared Distribution, Mathworld - Wolfram Web Resource, [7] Virtual Laboratory in Probability and Statistics, The Folded Normal Distribution, University of Alabama, 8

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