Born's rule and Schrödinger's equation

Question

In non-relativistic quantum mechanics, the equation of evolution of the quantum state is given by Schrödinger's equation and measurement of a state of particle is itself a physical process. Thus, should be governed by the Schrödinger's equation.

But we predict probabilities using Born's rule.

Do we use Born's rule just because it becomes mathematically cumbersome to account for all the degree of freedoms using the Schrödinger equation, so instead we turn to approximations like Born's rule.

So, is it possible to derive Born's rule using Schrödinger's equation?

Answer 1

Indeed, in non-relativistic quantum mechanics, the equation of evolution of the quantum state is given by Schrödinger's equation and measurement of a state of particle is itself a physical process and thus, should and is indeed be governed by the Schrödinger's equation.

Indeed, people like to predict probabilities using Born's rule, and sometimes they do this correctly, and sometimes incorrectly.

Do we use Born's rule just because it becomes mathematically cumbersome to account for all the degree of freedoms using the Schrödinger equation?

Yes and no. Indeed sometimes you can just use the Born rule to get the same answer as the correct answer you get from using the Schrödinger rule. And when you can do that, it is often much easier both computationally and for subjective reasons. However, that is not the reason people use the Born rule, they use it because they have trouble knowing how to relate experimental results to wavefunctions. And the Born rule does exactly that. You give it a wavefunction and from it you compute something that you know how to compare to the lab. And that is why people use it. Not the computational convenience.

Is it possible to derive Born's rule using Schrödinger's equation?

Yes, but to do so you need to overcome the exact reason people use the Born rule. All the Schrödinger equation does is tell us how wavefunctions evolve. It doesn't tell you how to relate that to experimental results. When a person learns how to do that, then they can see that the job done by Born's rule is already done by the unitary Schrödinger evolution.

How are probabilistic observations implied by causal evolution of the wave function?

The answer is so simple it will seem obvious. Just think about how you verify it in the lab, and then write down the appropriate system that models the actual laboratory setup, then setup the Schrödinger for that system.

For the Born rule you use one wavefunction for one copy of a system, then you pick an operator, and then you get a number between zero and one (that you interpret as relative frequency if you did many experiments on many copies of that one system). And you get a number for each eigenvalue in a way that depends on the one wavefucntion for one copy of a system even though you verify this result by taking a whole collection of identically prepared particles.

So that's what the Born rule does for you. It tells you about the relative frequency of different eigenvalues for a whole bunch of identically prepared systems, and so you verify it by making a whole bunch of identically prepared systems and measuring the relative frequency of different eigenvalues.

So how do you do this with the Schrödinger equation? Given the state and operator in question you find the Hamiltonian that describes the evolution corresponding to a measurement of the operator (as an example my other answer to this question cites an example where they explicitly tell you the Hamiltonian to measure the spin of a particle). Then you also write down the Hamiltonian for the device that can count how many times a particle was produced, and the device that write down the Hamiltonian for the device that can count how many times a particle was detected with a particular outcome, and the device that takes the ratio. Then you write down the Schrödinger equation for a factored wavefunction system that has a huge number of factors that are identically wavefunctions, and also where there are sufficient numbers of devices to split different eigenfunctions of the operator in question and the device that counts the number of results. You then evolve the wavefunction of the entire system according to the Schrödinger equation. When 1) the number of identical factors is large and 2) the devices the send different eigenfunction to different paths make the evolved eigenfunctions mutually orthogonal, then something happens. The part of the wavefunction describing the state of the device that took the ratio of how many got a particular eigenvalue evolves to have almost all of its $L^2$ norm concentrated over a state corresponding to the ratio that the Born rule predicts and is almost orthogonal to the parts corresponding to states the Born rule did not predict.

Some people will then apply the Born rule to this state of the aggregator, but then you have failed. We are almost there. Except all we have is a wavefunction with most of its $L^2$ norm concentrated over a region with an easily described state. The Born rule tells us that we can subjectively expect to personally experience this aggregate outcome, the Born rule says this happens with near certainty since almost all the $L^2$ norm corresponds to this state of the aggregator. The Schrödinger equation by itself does not tell us this.

But we had to interpret the Born rule as saying that those numbers between 0 and 1 correspond to observed frequencies. How can we interpret "the wavefuntion being highly concentrated over a state with an aggregator reading that same number" as corresponding to an observation?

This is literally the issue of the question, interpreting a mathematical result about a mathematical wavefunction as being about observations.

The answer is that we and everything else are described by the dynamics of a wavefunction, and that a part of a wave with small $L^2$ norm that is almost entirely orthogonal doesn't really affect the dynamics of the rest of the wave. We are the dynamics. People are processes, dynamical processes of subsystems. We are like the aggregator in that we are only sensitive to some aspects of some parts of the rest of the wavefunction. And we are robust in that we are systems that can act and time evolve in ways that can be insensitive to small deviations in our inputs, so the part of the wavefunction that corresponds to the aggregator having most of the $L^2$ norm concentrated on having the value predicted by the Born rule (ant that state with that concentration on that value is what the Schrödinger equation predicts) is something that can interact with us, the robust information processing system that also evolves according to the Schrödinger equation interacts with us in the exact same way as a state where all the $L^2$ norm was on that state, not just most of it.

This dynamical correlation between the state of the system (the aggregator) and us, the interaction of the two, is exactly what observation is. You have to use the Schrödinger equation to describe what an observation is to use the Schrödinger equation to predict the outcome of an observation. But you only need to do that on states very very very close to get the Born rule since the Born rule only predicts the outcomes of an aggregator's response to large numbers of identical systems. And those states are exactly the ones we can give a purely operational definition in terms of the Schrödinger equation.

We simply say that the Schrödinger equation describes the dynamics, including the dynamics of us, the things being "measured" and the whole universe. The way a measurement works is that you have a Hamiltonian that acts on your subsystem $|\Psi_i\rangle$ and your entire universe $|\Psi_i\rangle\otimes |U\rangle$ and evolves it like:

$$|\Psi_i\rangle\otimes |U\rangle\rightarrow|\Psi_i'\rangle\otimes |U_i\rangle.$$

The essential aspects of it being a measurement is that when $|\Psi_i\rangle$ and $|\Psi_j\rangle$ are in different eigenspaces they are originally orthogonal, but that orthogonality transfers over to $|U_i\rangle$ and $|U_i\rangle$ in such as way as to ensure the Schrödinger time evolution evolutions of $|\Psi_i'\rangle\otimes |U_i\rangle$ remain orthogonal. (And also we need that $|\Psi_i'\rangle$ is still in the eigenspace.) That's our restriction on the Hamiltonians that are used in the actual Schrödinger

What is the problem?

The problem is that we had to say how to relate a mathematical object to us and where probability words entered. And there isn't any probability. We just have ratios that look like the ratios that probability would predict for us if there were probabilities. And we have to bring up how our observations and experiences relate to the mathematics.

Historically there were strong objections to this, that talking about how human people dynamically evolve should not be relevant to physics. Seems like Philosophy the old fashioned objections would go. But if you think of people as dynamical information processors, then we can characterize them as a certain kind of computer that interacts with the wavefucntion of the rest of the world in a particular way. And other kinds of computer are possible, things we call quantum computers. And now we can make this excuse no longer. We need to talk about the difference between a classical computer that is designed to be robust against small quantum effects, and one that can be sensitive to these effects so that it can vontinue to interact before it has gotten to the point in the evolution where the Born rule could be used.

We must now own up to the fact that the Schrödinger equation evolution is the only one we've seen, and that is what corresponds to what we actually observe in the laboratory experiments where the Born rule is used. And we must own it so that we can correctly describe what happens in experiments where the Born rule doesn't apply, where as always we must use the Schrödinger equation.

Answer 2

There is a different interpretation of Schroedinger's equation than the one in terms of Born's rule or Copenhagen interpretation.

It is called Bohmian mechanics or deBroglie-Bohm-theory or pilot-wave theory. The general idea is to set the wave-function as $\Psi(t, \vec{x}) = R(t,\vec{x}) \cdot \exp(i\ \frac{S(t,\vec{x})}{\hbar})$, which is no restriction. Inserting this into Schroedinger's equation one gets two equations from the real and imaginary part. One of them is a conservation equation for a new charge-density $R^2$. The other is the classical Hamilton-Jacobi equation with an extra potential. The action is taken to be $S$. The new potential originates from the charge-density $R^2$.

If one then applies statistical mechanics to this classical theory, one sees that expectation values of a physical quantity $Q(\vec{x})$ are the same as they are in quantum theory, namely $<\Psi|Q|\Psi> = \int_\infty^\infty \Psi(t, \vec{x}) \ \ Q(\vec{x}) \ \ \Psi^*(t, \vec{x})\ \ d^3x$.

In a similar way, one can interpret Born's rule as the consequence of classical statistical mechanics with an extra force-law for the new charge $R^2$.

If you wanna look further into this, I suggest reading the original papers:

http://journals.aps.org/pr/abstract/10.1103/PhysRev.85.166

http://journals.aps.org/pr/abstract/10.1103/PhysRev.85.180

Answer 3

First I'll describe the (to me) cleanest and clearest example, then we'll extend it.

You have a wavepacket that in addition to a complex scalar field also specifies at every point a spin vector/plane. Since we want to describe a measurement we need a Hamiltonian that described the interaction with the device and the system, in our case the device is a Stern-Gerlach device, so the interaction Hamiltonian is the Hamiltonian that depends on magnetic moment (proportional to the spin vector orthogonal to the spin plane) and the inhomogeneity of the external magnetic field. When the spin vector points in the z-direction, the whole packet is deflected left. When the spin vector points the exact negative direction ( negative z-direction) the whole packet is deflected right. When the spin vector points in other directions some of the packet is deflected right and some of it is deflected left and the spin vector becomes polarized in the direction that portion of the wavepacket went. The size of the two packets is influenced entirely by how much that spin vector can be written as a combination of the basis spin states (set it up so that when the spin vector points in a direction $(x,y,z)$ the spin state is eigen to $x\sigma_x+y\sigma_y+z\sigma_z$ with eigenvalue 1, so for direction $(x,y,z)$ take the eigenvector $(a,b)$ of $x\sigma_x+y\sigma_y+z\sigma_z$ with eigenvalue 1 and then $|a|^2:|b|^2$ is the proportion of the sizes of the wavepackets). All of this is predicted by the Schrödinger equation. See this nice article in the American Journal of Physics, (arxiv version).

If we set up our processes so that those two wavepackets will never again cross and interfere (such as by so called decoherence with the environment) then a remarkable simplification occurs. By linearity each acts on its own, and because of the never-again overlap, the sum of the integral of the square of the wavefunctions equals the integral of the square of the sum of the wavefucntions, this is a property of orthogonal functions, and they are forever orthogonal. So each can mathematically act like the other doesn't exist, and the whole universe is a combination of this part and the rest of the universe, so the whole universe is now and forever the sum of two orthogonal parts. Mathematically each can act like the other doesn't exist. And if we want to use the wavefunction to make our predictions, our predictions can act like the other option (other wavepacket) doesn't exist. At this point (or any later point) you might pretend a collapse happened and no one can gainsay you because each possibility now acts like it is the only thing that happened.

OK. So that's what is happening mathematically. What happens when we do the experiment? We are part of the universe and the wavefunction for everything can be written as a sum of each orthogonal wavepacket. Each acting like the other one doesn't exist, so we have the potential to be something that interacts with one wavepacket or the other, so we can interact with the wave as if it is either one that went left or one that went right. So we can talk about it like it went one way or the other. So we can do that over and over again.

Subjectively (i.e. experimentally) we very often noticed that the proportion of times we saw particular results is very close to being proportional to the integral of the squared length of the wavepacket. (With a deviation consistent with experimental noise and the kinds of effects seen whenever taking a small, rather than a large, sample from a probability distribution.)

Born worked with reasoning about scattering (into angles) rather than into left versus right, but the same thing really.

You could leave it at that. The math of the Schrödinger equation predicts branching of the wavepacket into orthogonal parts, and predicts various ratios of the integral of squared length and you could choose to only say that those ratios matches observed relative frequencies. Or you can try to go farther (as the Born rule does) and try to say that the squared length is an actual probability density. If you do that there are some serious problems.

Number one, you are assigning probabilities not to actual experimental outcomes, but assigning probability densities to regions of space and time in which no experimental action is being done (we can't say an observation has happened until the wavefunction develops orthogonal parts that will always be orthogonal forevermore, before that all the wavefunction does is evolve). This is storytelling, not science. Which is fine as long as it doesn't pretend to be science. However there is a problem with the storytelling. As explained in Lost Causes in Physics by R. F. Streater, if you assume a sample space (the mathematical underpinning of the mathematical theory of probability) then you can't handle random variables for noncommuting observables. If you select actual experiments to do you can select a maximal commuting algebra of observables and then make a sample space and then get a probability theory. But once you've done that, you get actual branching and are back to the special case. So the implicit assumption that probabilities make sense even when talking about a situation where measurements aren't happening is actually flawed completely because a probability requires a sample space and it is premature to have a probability theory (at least in the way mathematicians have done it, we could try to make a totally new probability theory from scratch, and while that might be good for science it is dishonest to call something a probability if you then have to discard all of existing probability theory and make a completely new probability theory from scratch just to call something a probability).

So it's actually wrong to think of the squared length of the wavefunction as a probability density. However it is fine to think of the integrals of the squares of mutually orthogonal sets of wavefunctions, and if they are forevermore orthogonal, then they can act as if in a universe alone and the relative integrals can be relative frequencies. And not only can they be, then that actually agrees with observations.

Answer 4

Derivations of the Born rule have been proposed, but they have all been criticized for invoking circular reasoning as CuriosOne mentioned in the comments. You can read a review of the arguments by Huw Price here. Zurek has invoked the fact that due to decoherence, observers are always competely entangled with the environment and then you can reason based on certain symmetries that completely entangled states will be subject to, see here for details. But by bringing in the environment, he is hiding a circular form of reasoning.

What has been proven is a derivation from a weaker statement that says that if a system is in an eigenstate of an observable, then measuring that observable will yield the eigenvalue corresponding to that eigenstate with certainty.

The proof is rather trivial, it follows from considering making N observations on N identically prepared systems using a hypothetical device that remains in quantum coherence. The state of the device will then contain all the information of the measurement outcomes, including how large the deviation from Born's rule is. You can then construct an observable that corresponds to measuring this deviation. In the limit of N to infinity, you can show that the state of the device will converge to the null space of that observable. Therefore, by the weaker postulate, you find that Born's rule is satisfied.

Answer 5

Your question is misconceived. The wavefunction does not collapse. Rather, when you measure a system, each of the possible outcomes of that measurement happens. Each outcome is associated with a different version of the measurement apparatus. Those different versions of the measurement apparatus can't interact with one another or exchange information. The way in which the wavefunction is sliced up into different versions is a result of the fact that information can only be copied from one system to another if it is instantiated in a set of mutually orthogonal projectors:

http://arxiv.org/abs/1212.3245.

Quantum mechanics isn't a stochastic theory. The probability of a measurement outcome doesn't refer to the chances of picking it out of a hat. Rather, the Born rule is a measure over the set of outcomes of a measurement that satisfies the constraints imposed by decision theory:

http://arxiv.org/abs/quant-ph/9906015.

If you wanted to bet on the outcome of a quantum mechanical experiment the way to do this that would make you money would be to use the Born rule to calculate the amount you expect to make.

Answer 6

According to the Many Worlds Interpretation, the measurement device, and by extension everyone who reads it, gets entangled with the particle in exactly the way the Schrödinger equation predicts. The Born rule tells you how subjectively likely you are to be in a given universe. It's basically just a really bizarre anthropic principle. Feel free to interpret that as MWI being generally right, and something we don't have enough information to guess at explains the Born rule using more sensible anthropics.

According to the Copenhagen interpretation, when too many particles get entangled, the Schrödinger equation momentarily stops applying and the waveform collapses into one of the possible states with the probability specified by the Born rule.

The reason the Schrödinger equation and the Born rule don't look like they fit together well is that they don't fit together well.

Answer 7

Thus, should be governed by the Schrodinger's equation.

Schroedinger's equation is useful for description of atomic systems. Measurement of such systems involves macroscopic bodies. It is out of place to apply Schroedinger's equation to these macroscopic bodies; classical description is simpler, more accurate and more useful. Why would you describe position of a needle in an ammeter by a complex function of zillion variables, when all it would give is probability density and in practice, one real number - angle - is sufficient?

But it becomes mathematically cumbersome, so we go to approximations like Born's rule.

Born's rule is not an approximation to something deeper. It was proposed by Born as a simple and hopeful rule to understand the $\psi$ function Schroedinger introduced with his equations. Born introduced his rule for description of scattering experiments and later it got generalized to be the cornerstone of interpretation of $\psi$ in any Schroedinger equation. It gives probability density in configuration space of the system described by $\psi$. Without the rule, the $\psi$ function would be merely a by-product of calculations seeking the Hamiltonian eigenvalues; with it, many phenomena (oscillation of electrons...) can be probabilistically described with $\psi$.

How do we unite Born's probabilistic collapse with the Schrodinger's equation evolution?

The Born rule does not imply any collapse. When we have function of three coordinates $\psi(x,y,z)$ for electron in hydrogen atom, the Born rule says $|\psi|^2$ is probability density for position of this point electron.

The electron is a point, the wave function is not. The electron is not the same thing as the wave function that describes it, just as pollen grain is not the gaussian function often used to describe it.

Wave function is a mathematical function and can be applied even to systems of many particles, such as the water molecule. Again, water molecule is not a wave function of many variables; the latter describes the water molecule.

The wave function never collapses to a point - there is no normalizable function that would be concentrated in a point. That is no problem for applicability of $\psi$, since no measurement of position (configuration) is exact; some inacurracy is always present.

So the appropriate thing to do after position is registered is to change the wave function into the best function compatible with the registered position, taking into account the measure of uncertainty the measurement is characterized with.

This procedure is not a result of the Schroedinger equation, because it uses the result of measurement and information on its accuracy, which is never determined by the Schroedinger equation.

The Schroedinger equation is limited. It contains no relativity and it is merely a probabilistic description. It cannot be used to predict the results of actual events; it only gives probabilities of results given probabilities of initial conditions. That is, when it is used according to the Born rule.