B What is the origin of Observables?

[entropy1]

Why does an Observable have to be Hermitian, and why do the eigenstates and eigenvalues have to respresent the possible measured values? Is is by definition? What is the origin of this convention?

 

[A. Neumaier]

a

As the discussion here at https://www.physicsforums.com/posts/5513644 shows, observables don't have to be Hermitian and often aren't. Only those to be measured by an ideal von-Neumann measurement must be Hermitian, because this guarantees the existence of the projections required by Born's rule.

The origin of Born's rule are experimental observations that suggested the rule as being valid for certain simple, paradigmatic experiments. In general one needs POVMs to model realistic measurements.

 

[Mentz114]

Why does an Observable have to be Hermitian, and why do the eigenstates and eigenvalues have to respresent the possible measured values? Is is by definition? What is the origin of this convention?

Introduced by fiat in chapter II of PAM Dirac, The Principles of Quantum Mechanics.

You can read it here https://archive.org/details/DiracPrinciplesOfQuantumMechanics

 

[Nugatory]

Why does an Observable have to be Hermitian?

Only Hermitian operators are guaranteed to have only real eigenvalues... and it seems a reasonable enough postulate that any measured value must be a real number.

and why do the eigenstates and eigenvalues have to represent the possible measured values? Is is by definition? What is the origin of this convention?

That's another postulate. We use these postulates because they work; that is, the mathematics that follows from them makes accurate quantitative predictions about how the universe behaves. You might find wikipedia's article on the history of the mathematical formalism interesting: https://en.wikipedia.org/wiki/Mathematical_formulation_of_quantum_mechanics#History_of_the_formalism

 

[A. Neumaier]

any measured value must be a real number.

The frequency of a damped harmonic oscillator is a measurable, complex number. It occurs over and over in the electrodynamics of circuits.

 

[Physics Footnotes]

Although the ultimate answer to "Why is this physical theory the way it is?" will always be "Because it works.", we shouldn't give, or be satisfied with, that answer to every question pertaining to the given theory.

We can do much better than that by identifying the crucial underpinning assumptions of the theory, mandating them as unexplained edicts, and then showing how the rest of the theory can be obtained through a process of deductive reasoning (sometimes rigorous theorem-proving, other times plausible physical arguments).

In the case of Quantum Mechanics, I think a good starting point is to accept that a system is to be modeled by a complex Hilbert Space $\mathscr{H}$, and that experimental outcomes are to be interpreted as probability distributions. The heart and soul of the prescription is that probabilities are ultimately calculated in terms of expressions of the form $|\langle \psi | \phi \rangle |^2$, where $\psi$ and $\phi$ are unit vectors in $\mathscr{H}$.

It is useful to think of the 'bra' $\langle \psi |$ as the simplest possible measurement procedure (or observable) and the 'ket' $| \phi \rangle$ as the simplest possible preparation procedure (or state).

The rest of the theory (i.e. this and that type of operator, trace functionals, and all that jazz) emerges from fleshing out this simple prescription into a well-defined and self-consistent probability theory.

Without going into all the details here (you can read my more detailed explanation here if you're interested), the Hermitian Operators (more correctly Self-Adjoint Operators) are a natural generalization of the 'observable' side of the recipe, which assign real numbers to outcomes while preserving the basic probability calculus. Similarly, Density Operators are a natural way to generalize the 'state' side of the recipe by allowing us to consider 'convex combinations' of states.

When looked at this way we can come up with further ways to generalize the recipe...

From the fact that a self-adjoint operator is equivalent to a Projection-Valued Measure (PVM), for example, we can generalize to things called Positive Operator Valued Measures (POVMs) whilst keeping the probability calculus intact.

In fact, we can generalize in another direction too. We can attach complex numbers to measurement values, instead of real numbers, and the resulting object is a Normal Operator $N$ (which has the form $N=Ae^{i\Theta}$ for self-adjoint operators $A$ and $\Theta$ ).

And on it goes...

The main requirement in Quantum Mechanics is not that measurement values are real, or that outcomes must be eigenvalues, or that post-measurement states must be eigenvectors, or any of those things. They were just part of the historical discovery pathway. Rather the aim of the game is to make sure you have an unambiguous and self-consistent probability calculus which preserves the basic interpretation of $|\langle \psi | \phi \rangle |^2$ as the probability of a simple observable $\langle \psi |$ registering an outcome for the pure state $| \phi \rangle$.

 

[MichPod1]

One may start with a postulate that for a particular observable the states for which it has uniquly defined measured value $a_i$ are orthogonal: $\langle\psi_i|\psi_j\rangle=0$ for $i\neq j$
Without loosing of generality, we can also normalize them and further consider them orthonormal. $\langle\psi_i|\psi_j\rangle=\delta_{ij}$

Then the operator of the observable is introduced as $\hat A = \sum_{i} a_i|\psi_i\rangle\langle\psi_i|$

It may be easily seen that this operator has $|\psi_i\rangle$ as eigenvectors and $a_i$ as respective eigenvalues. Also, it is seen that if $a_i$ are real, then $\hat A$ is hermitian.

 

[Truecrimson]

One may start with a postulate that for a particular observable the states for which it has uniquly defined measured value $a_i$ are orthogonal: $\langle\psi_i|\psi_j\rangle=0$ for $i\neq j$
Without loosing of generality, we can also normalize them and further consider them orthonormal. $\langle\psi_i|\psi_j\rangle=\delta_{ij}$

To give a slightly different perspective, when we calculate the average with a density operator $\rho$, we multiply $\text{Tr}(\rho |\psi_i \rangle \langle \psi_i |)$ with the value $a_i$ associated to the outcome $| \psi_i \rangle$ and add them all up. What do we get? $$ \sum_i a_i \text{Tr}(\rho |\psi_i \rangle \langle \psi_i |) = \text{Tr} \left( \rho \sum_i a_i |\psi_i \rangle \langle \psi_i | \right) $$ So arranging eigenvectors and eigenvalues into a Hermitian operator can be thought of as merely a convenient way to calculate an average.

 

[A. Neumaier]

So arranging eigenvectors and eigenvalues into a Hermitian operator can be thought of as merely a convenient way to calculate an average.

Yes, and allowing in place of the rank 1 projectors arbitrary positive definite Hermitian operators leads in this way naturally to POVMs. This shows that the latter are far more natural. The emphasis on the special case of rank 1 projectors (leading to Born's rule) is a historical accident only.

 

[entropy1]

I was thinking, do we use Hermitian Operators because they have eigenvectors, and eigenvectors are compatible with the phenomenon of collapse that we observe?

 

[A. Neumaier]

I was thinking, do we use Hermitian Operators because they have eigenvectors, and eigenvectors are compatible with the phenomenon of collapse that we observe?

In general, collapse is not to an eigenstate since observations are almost never perfect. Thus this gives no argument in favor of Hermitian operators.

The only reason why some Hermitian (more precisely self-adjoint) operators have a distinguished meaning is that they are generators of unitary transformations. In many cases, these unitary transformatins have a kinematical meaning as symmetries or dynamical symmetries. They form the bridge to the group theoretical aspects of quantum mechanics.

 

[Nugatory]

I was thinking, do we use Hermitian Operators because they have eigenvectors

No. Operators don't need to be Hermitian to have eigenvectors.

and eigenvectors are compatible with the phenomenon of collapse that we observe?

No, because collapse is not a phenomenon that we observe.

There is no substitute for learning what words like "Hermitian" and "eigenvector" mean before you start using them.

 

[entropy1]

because collapse is not a phenomenon that we observe.

I learned that when you measure, for instance, electron spin, getting as a result (for instance) 'up spin' has as a consequence that subsequent measurements along the same axis gives a 100% probability of yielding the spin along that same axis. Are you saying that we haven't observed collapsing the state of the spin there?

 

[Nugatory]

Yes, that

I learned that when you measure, for instance, electron spin, getting as a result (for instance) 'up spin' has as a consequence that subsequent measurements along the same axis gives a 100% probability of yielding the spin along that same axis.

Yes, this follows from the mathetatical formalism of quantum mechanics. But...

Are you saying that we haven't observed collapsing the state of the spin here?

Yes, that is what I and other people other people have been saying, for more posts now than I care to count.

Collapse is not part of quantum mechanics. It is one of many interpretations, metaphors that we use to help us form a mental picture of what the equations are saying. The only reason to adopt any interpretation is because it helps you think about the problem at hand... and if it's not doing that you shoudl forget about it.

 

[A. Neumaier]

when you measure, for instance, electron spin, getting as a result (for instance) 'up spin' has as a consequence that subsequent measurements along the same axis

How would you do this? Single particle states are very fragile.

One cannot measure the spin of an electron getting a definite result without having lost the electron in the detector. So one can't do a subsequent measurement on the electron.

All one can do is prepare an electron state in a particular way so that its state is known by preparation although it hasn't been measured, and then measure it once to check the validity of the preparation or the validity of quantum mechanics.

 

[bolbteppa]

If you take the Heisenberg Uncertainty Principle as Landau-Lifshitz state it, saying 'there is no concept of the path of a particle', motivated by the inability of measuring the path of an electron path through a cloud, then since a path is specified by position & velocity and initial conditions, having measured the positions at each point you measure at implies the velocities f(i) are randomly distributed & that all we can do is find an average velocity along the path. This motivates using the probabilistic notion of 'expected value' to formalize finding the average/expected velocity of, say, n measurements. Then a simple calculation motivates, taken from Parthasarathy's QM book (page 1), different formalisms of quantum mechanics:

The final equality illustrates why you have Hermitian operators acting on vectors, the middle line illustrates the density matrix formalism, the top line illustrates expectation as an inner product.

 

[Traruh Synred]

The frequency of a damped harmonic oscillator is a measurable, complex number. It occurs over and over in the electrodynamics of circuits.

https://skepticalsciencereviews.wordpress.com/reviews/

See "Are imaginary numbers real" at above site.

They are! They are two numbers. We measure 'em all the time in interference experiments. They are just as real as a tensor describing Electromagnetic fields. Nothing 'imaginary' about 'em.

 

[Derek Potter]

The frequency of a damped harmonic oscillator is a measurable, complex number. It occurs over and over in the electrodynamics of circuits.

It's not measurable as a complex number though.

What we actually have is a typically a second order linear circuit modeled by a couple of linear equations. The general solutions are then complex exponentials. But physics constrains the actual solutions to occur in superposition - the familiar expansion of a sine or cosine as the sum of two imaginary exponentials.

So whilst you can correctly say the natural frequency is complex, the physical observables still have to be real.

 

[Traruh Synred]

It's not measurable as a complex number though.

What we actually have is a typically a second order linear circuit modeled by a couple of linear equations. The general solutions are then complex exponentials. But physics constrains the actual solutions to occur in superposition - the familiar expansion of a sine or cosine as the sum of two imaginary exponentials.

So whilst you can correctly say the natural frequency is complex, the physical observables still have to be real.

 

[Traruh Synred]

A complex number is just two real numbers with a rule for 'multiplying' them. A simple rate is just one real number. If it varies in time or space it takes more than one real number to describe the dependence. The measurements of dependence are just as real as anything. There's in fact even then no need to use 'complex' or 'imaginary' numbers, but that notation gives a more elegant formulation. It's not different than tensors or matrices in other context. Just notation, but good notation.

 

[Derek Potter]

A complex number is just two real numbers with a rule for 'multiplying' them. A simple rate is just one real number. If it varies in time or space it takes more than one real number to describe the dependence. The measurements of dependence are just as real as anything. There's in fact even then no need to use 'complex' or 'imaginary' numbers, but that notation gives a more elegant formulation. It's not different than tensors or matrices in other context. Just notation, but good notation.

Yes I know what a complex number is.

But we are not talking about whether complex numbers are handy. We are talking about physical properties necessarily being real. A. Neumaier gave complex frequency as a counter-example. For it to be a valid counter-example it would have to be complex-valued, which it is, and physically possible, which, in general, complex frequencies are not. That is to say, physically possible solutions are a subspace of the complex solution space. (They are confined to solutions that are real in the time domain.) Do you see what I'm saying?

 

[A. Neumaier]

l. A. Neumaier gave complex frequency as a counter-example. For it to be a valid counter-example it would have to be complex-valued, which it is, and physically possible, which, in general, complex frequencies are not.

A damped harmonic oscillator has a physically meaningful complex frequency.

 

[Derek Potter]

A damped harmonic oscillator has a physically meaningful complex frequency.

Of course, but we are not talking about whether you can assign a meaning to it. We are talking about whether it represents an observable quantity. Let's be clear, this is not an abstract philosophical or semantic point. A function such as C.exp(iω-k)t is complex-valued and cannot represent anything observable. You must interpret the solution by projecting it onto the real axis etc.

 

[Traruh Synred]

Yes I know what a complex number is.

But we are not talking about whether complex numbers are handy. We are talking about physical properties necessarily being real. A. Neumaier gave complex frequency as a counter-example. For it to be a valid counter-example it would have to be complex-valued, which it is, and physically possible, which, in general, complex frequencies are not. That is to say, physically possible solutions are a subspace of the complex solution space. (They are confined to solutions that are real in the time domain.) Do you see what I'm saying?

 

[Traruh Synred]

An 'imaginary' frequency is just a plain old exponential or a 'sinh' or something. Nothing unreal about it. It's just words. A frequency could be called an 'imaginary' value of a a decay or growth rate. All the numbers whether in pairs or not are, of course, numbers.

 

[Derek Potter]

An 'imaginary' frequency is just a plain old exponential or a 'sinh' or something. Nothing unreal about it. It's just words. A frequency could be called an 'imaginary' value of a a decay or growth rate. All the numbers whether in pairs or not are, of course, numbers.

You can't have it both ways. If you want to say that a complex frequency is actually a real-valued function then it isn't an example of a complex-valued observable.

 

[Traruh Synred]

Sure I can have it both ways. It's just words. Complex observable is just a word for needing two numbers to describe the measurement. It get's elegant when the multiplication is defined to form a 'ring' . I can _call _that measuring a complex variable. In the other sense all the numbers that get measured a 'real' whether call them 'imaginary part' or not. For a rate at a single time you only need one number to specify it.

 

[Derek Potter]

Sure I can have it both ways. It's just words. Complex observable is just a word for needing two numbers to describe the measurement. It get's elegant when the multiplication is defined to form a 'ring' . I can _call _that measuring a complex variable. In the other sense all the numbers that get measured a 'real' whether call them 'imaginary part' or not. For a rate at a single time you only need one number to specify it.

Well, I'll try one more time.

Of course I agree that you can shuffle the real and imaginary names around, in fact the very idea of imaginary frequency for real-valued exponentials only arises because of the convention of excluding the √-1 from the frequency: exp(iωt) has an angular frequency of ω, not iω. Thus exp(-t/τ) must be assigned a frequency of i/τ. But this is nothing to do with whether you can observe a complex frequency. There are no observable "frequencies" rattling around in a circuit, real or complex.. You have to observe, say, voltage, as a function of time and apply a Laplace transform. In that way you represent a real-valued observable by a complex number. That's fine, you can represent it as a cat if you want to, the observable is still a real-valued quantity.

 

[A. Neumaier]

you represent a real-valued observable by a complex number.

In a damped harmonic oscillator, the complex-valued frequency has a nonzero real and imaginary part. The real observables are the points on the oscillating curve; the observable complex frequency is extracted from these and produces the physical way of summarizing the behavior of the oscillator.

It is always the summary that carries the physics. Without summarizing what happens in Nature we cannot form a single concept. Every observable is an abstraction of the real thing, and as an abstraction it may be a real number, a complex number, or an even more complicated object such as a vector or a tensor.

 

[Derek Potter]

In a damped harmonic oscillator, the complex-valued frequency has a nonzero real and imaginary part.

Well, I'll try one more time.

 

[Derek Potter]

In a damped harmonic oscillator, the complex-valued frequency has a nonzero real and imaginary part.

The real observables are the points on the oscillating curve; the observable complex frequency is extracted from these and produces the physical way of summarizing the behavior of the oscillator.

It is always the summary that carries the physics. Without summarizing what happens in Nature we cannot form a single concept. Every observable is an abstraction of the real thing, and as an abstraction it may be a real number, a complex number, or an even more complicated object such as a vector or a tensor.

Nice post-editing :)

Sure, that's what I said, the real variable is a projection of the complex representation.

I am surprised that nobody else seems to be interested in whether the idea of complex observables makes sense physically. You and I are clearly not going to agree about that so we must agree to differ.

 

[forcefield]

I am surprised that nobody else seems to be interested in whether the idea of complex observables makes sense physically.

Take a wire and bend it to a sine wave :)