Index Notation & Dirac Notation

[TheAustrian]

No. The measurement outcome will be an eigenvalue of the measurement operator. The state after the measurement will be the eigenstate corresponding to the eigenvalue that was the outcome.

The Schroedinger equation tells you how the state evolves between measurements.



Quantum mechanics only predicts probabilities or expectation values. An expectation value is simply an "average". Statistical mechanics and Mendelian genetics are two other theories that only give you probabilities or averages. These theories are all useful, even though they only predict expectation values, ie. average quantities.



Think about statistical mechanics. There every physical quantity calculated is an expectation value. It is an average. It tells you the average value if you do the measurement many times. In the same way that statistical mechanics is a useful theory although it only makes predictions about averaged quantities, quantum mechanics is also useful for making predictions about averages. Of course, it is you that has to choose which quantities you are interested in measuring, and quantum mechanics will give you the answer about their average values.

So expectation values are not particularly important? Wavefunctions, eigenstates and eigenvalues are the important stuff?

 

[bhobba]

This might be a stupid question, but what exactly is a wave-function collapse? Does it have something to do with expectation values? When I calculate an expectation value, is that some probability that I will obtain that value at a wavefunction collapse or its something else? I think I understood parts (1) and (2).

Ballentine explains it. Enough said.

When you have gone through it we can have a real nifty chat about it and something called Gleasons theroem. It's associated with being able to derive Borns rule from that single axiom I mentioned to Atyy

Thanks
Bill

 

[atyy]

So expectation values are not particularly important? Wavefunctions, eigenstates and eigenvalues are the important stuff?

Expectation values, which are just another name for average values, are very important. Quantum mechanics only predicts probabilities. Equivalently, quantum mechanics only predicts expectation values or averages.

Let's say you measure an observable that corresponds to an operator which has only 2 eigenvalues 1 and -1. This means that the outcome of a measurement can only be either 1 or -1. Let's say that quantum mechanics predicts that 60% of the time you get 1, and 40% of the time you get -1. The expectation value or average value of the outcomes for this measurement is (0.6)(1) + (0.4)(-1) = 0.2.

 

[kith]

It's pretty obvious when you think about the archetypical observation he talks about. Nothing mysterious about it - he simply leaves it up in the air as being utterly obvious the apparatus is classical.

This is not correct. His quantum treatment of the apparatus leads to equation (9.8) about which he says "This final state is a coherent superposition of macroscopically distinct indicator eigenvectors [...]" and "In a typical case, the indicator variable α_r might be the position of a needle on a meter or a mark on a chart recorder, and for two adjacent values of the measured variable, r and r , the separation between α_r and α_r could be several centimeters." and concludes "Each member system of the ensemble consists of an object and a measuring apparatus."

The macroscopic superpositions in the measurement process are the very reason why he takes the ensemble viewpoint.

 

[Nugatory]

Let's say you measure an observable that corresponds to an operator which has only 2 eigenvalues 1 and -1. This means that the outcome of a measurement can only be either 1 or -1. Let's say that quantum mechanics predicts that 60% of the time you get 1, and 40% of the time you get -1. The expectation value or average value of the outcomes for this measurement is (0.6)(1) + (0.4)(-1) = 0.2.

Of course in this example the expectation value is not as interesting as the probability distribution from which it is calculated - the average American house may have 2.8 bedrooms, but I've never seen a house with a fractional bedroom. When the spectrum is continuous (as it is for the position of an unbound particle, for example) the expectation value becomes a much more useful quantity.

(I'm not disagreeing with atyy here, just extending his answer to TheAustrian's question about whether the expectation values or or the eigenvalues are "the important stuff").

 

[atyy]

Of course in this example the expectation value is not as interesting as the probability distribution from which it is calculated - the average American house may have 2.8 bedrooms, but I've never seen a house with a fractional bedroom. When the spectrum is continuous (as it is for the position of an unbound particle, for example) the expectation value becomes a much more useful quantity.

(I'm not disagreeing with atyy here, just extending his answer to TheAustrian's question about whether the expectation values or or the eigenvalues are "the important stuff").

Yes. I left this point out for simplicity. One could choose to say that the probability distribution of outcomes, ie. the probability distribution of eigenvalues, is more fundamental. It's a matter of taste, because for physically reasonable distributions, the probability distribution and a full set of certain expectation values called cumulants are equivalent. The cumulants are a set of expectation values that can be generated by Taylor series, and whose low order terms are the mean, variance, skew, kurtosis etc.

 

[WannabeNewton]

So expectation values are not particularly important? Wavefunctions, eigenstates and eigenvalues are the important stuff?

I understand where your state of distraught comes from. In statistical mechanics we take the partition function of a system at equilibrium with an external heat bath and calculate quantities like average pressure, magnetic susceptibility, average energy etc. which all clearly have very important physical applications and are put to use extensively everyday in calculations.

In QM, if I have a pure state, what use are the expectation values of all the observables relevant to the system? What can I actually do with these expectation values? Sure I can calculate them at whim but can I put actually them to use in other calculations of physical applications? Do they hold the same all-important place that they do in statistical thermodynamics? You probably remember them showing up in non-degenerate and degenerate first order time-independent perturbation theory but where else do they show up?

Are these questions more representative of your own?

 

[Fredrik]

So expectation values are not particularly important? Wavefunctions, eigenstates and eigenvalues are the important stuff?

It's at least as important as all the things you mentioned. As I said in post #82, the assumption about the significance of expectation values is essentially what turns the mathematics of QM into physics.

Expectation values, which are just another name for average values,

I have to object to this part. The average value and the expectation value are certainly not defined the same way. That they have the same value is an extremely important assumption, an assumption that's part of the definition of QM.

Same thing with the probability of a possible result and the relative frequency of that result. If you want, you can replace the assumption that expectation values are equal to average values with an assumption that says that probabilities are equal to relative frequencies. These two correspondence rules are essentially equivalent.

There's a similar thing in SR. It always bugs me when people say that proper time is what clocks measure. It's not, at least not by definition. Proper time is the number you get when you integrate a certain function along a timelike curve. That this is equal to what clocks measure is a major assumption. It's the most important correspondence rule in both SR and GR. It's part of the definitions of those theories.

 

[atyy]

I have to object to this part. The average value and the expectation value are certainly not defined the same way. That they have the same value is an extremely important assumption, an assumption that's part of the definition of QM.

Same thing with the probability of a possible result and the relative frequency of that result. If you want, you can replace the assumption that expectation values are equal to average values with an assumption that says that probabilities are equal to relative frequencies. These two correspondence rules are essentially equivalent.

I don't agree. The expectation and the average value are the same by definition in probability. One is the formal term, the other is the informal term.

I do agree that it is an additional assumption to say that probability is operationally defined as relative frequency.

 

[stevendaryl]

I don't agree. The expectation and the average value are the same by definition in probability. One is the formal term, the other is the informal term.

I do agree that it is an additional assumption to say that probability is operationally defined as relative frequency.

Well, whatever it is that you want to call the expression \langle \Phi|O|\Phi \rangle, it's an assumption that it's equal to the average value of observable O in the state described by |\Phi\rangle.

 

[atyy]

Well, whatever it is that you want to call the expression \langle \Phi|O|\Phi \rangle, it's an assumption that it's equal to the average value of observable O in the state described by |\Phi\rangle.

But it is not an assumption different from the assumption that it is the expectation value of the observable.

 

[Fredrik]

I don't agree. The expectation and the average value are the same by definition in probability. One is the formal term, the other is the informal term.

Nitpicking a bit: Both have formal definitions, and they're different. There are theorems ("laws of large numbers") that prove that the average converges in at least two different ways to the expectation value.

I was ignoring that there's a formal definition of "average". My point was that regardless of what mathematical terms we define, the physics is in the assumptions that tell us how the math is related to the real world. I consider this a supremely important fact; it's the very foundation of the philosophy of mathematics and science. Because of this, I find it rather odd that pretty much everyone but me (definitely not just you) are going out of their way to avoid mentioning these correspondence rules explicitly. Instead they seem to prefer to connect mathematics to reality simply by using the same term for the real-world thing and the corresponding mathematical thing. This is what people do with "proper time" in the relativity forum. I don't like it because it hides what we're really doing.

Edit: Even though "average" and "expectation value" are essentially the same in pure mathematics (because of the laws of large numbers), there's a difference between these concepts and the real-world average, i.e. the average of the measurement results in an actual experiment. The assumption that the real-world average is equal to a number given by a mathematical formula, is a fundamental assumption. This correspondence rule is a hugely important part of the definition of the theory.

I do agree that it is an additional assumption to say that probability is operationally defined as relative frequency.

Probability is typically defined something like this: Let $X=\{1,2,3,4,5,6\}$. I'll use the notation $|E|$ for the number of elements of $E$. For each $E\subseteq X$, define $P(E)=|E|/|X|$. For each $E\subseteq X$, the number $P(E)$ is called the probability of E. Note that there's no connection to the real world whatsoever. The probability measure P is just an assignment of numbers in the interval [0,1] to subsets of some set X. Now we can turn this piece of pure mathematics into a falsifiable theory about the real world with a few simple assumptions. Some of them are common to all probability theories. But at least one assumption is part of the definition of the specific probability theory. In our case, that assumption can be that the elements of the set X correspond to the possible results of a roll of a six-sided die.

Well, whatever it is that you want to call the expression \langle \Phi|O|\Phi \rangle, it's an assumption that it's equal to the average value of observable O in the state described by |\Phi\rangle.

But it is not an assumption different from the assumption that it is the expectation value of the observable.

If "average" refers to the average of the results of actual measurements, then I would say that it's definitely a different assumption. Not only that. It's a very different kind of assumption.

There's a mathematical definition that associates the term "expectation value" with that number. There's a mathematical definition that associates the term "average" with something else. There are theorems that tell us how the average converges to the expectation value. But all of this is pure mathematics. The physics is in the correspondence rules, not in the mathematics, and in this case, the correspondence rule is the assumption that the expectation value is equal to the real-world average (not just the "formal average", i.e. the number that the laws of large numbers are making claims about).

 

[atyy]

Nitpicking a bit: Both have formal definitions, and they're different. There are theorems ("laws of large numbers") that prove that the average converges in at least two different ways to the expectation value.

I was ignoring that there's a formal definition of "average". My point was that regardless of what mathematical terms we define, the physics is in the assumptions that tell us how the math is related to the real world. I consider this a supremely important fact; it's the very foundation of the philosophy of mathematics and science. Because of this, I find it rather odd that pretty much everyone but me (definitely not just you) are going out of their way to avoid mentioning these correspondence rules explicitly. Instead they seem to prefer to connect mathematics to reality simply by using the same term for the real-world thing and the corresponding mathematical thing. This is what people do with "proper time" in the relativity forum. I don't like it because it hides what we're really doing.

I agree with your point, but not your nitpicking. The link you give always uses the term "sample average" in the theorems, not the unqualified "average". The main quirk with my terminology is that the unqualified "average" most usually means "mean", which is simply a particular expectation value, ie. the first cumulant. However, I meant it as a synonym for "expectation value". For example, in my colloquial way of using the word "average", the variance is the average of the squared deviation from the mean. Basically, in all my statements I am assuming (1) Kolmogorov axioms (2) relative frequency interpretation of probability, but that is even before quantum mechanics, ie. I assume it for Mendelian genetics and classical statistical mechanics. My assumption (2) is the same as what you are calling the "correspondence" between maths and physics.

 

[bhobba]

Nitpicking a bit: Both have formal definitions, and they're different. There are theorems ("laws of large numbers") that prove that the average converges in at least two different ways to the expectation value.

True. But they are in probability (the weak law) and almost surely (the strong law). To apply them you are going to have to make some assumption such as for all practical purposes if a probability is infinitesimally close to one you can take it as a dead cert. People tend to do that sort of thing unconsciously without even realizing it, such is the very intuitive understanding we all have of probability.

My point was that regardless of what mathematical terms we define, the physics is in the assumptions that tell us how the math is related to the real world. I consider this a supremely important fact; it's the very foundation of the philosophy of mathematics and science.

You are correct.

But in applied math, especially with regard to probability, people generally aren't that careful and you make all sorts of unconscious assumptions you generally aren't even aware of when applying it.

That's the reason QM can seem like applied math rather than physics. Its we have this very intuitive understanding of probability that gets us by.

I think most physicists, unlike applied math guys, haven't even done an advanced course on probability and statistical modelling and get by just fine. Pure math guys - that's another story - there are very subtle issues associated with a rigorous approach to probability such as the proof a Wiener process actually exists, is continuous but nowhere differentiable.

In QM as far as an axiomatic treatment is concerned I use Kolmogorov's axioms. Each interpretation, in applying it, has a slightly different take on how its done eg the ensemble interpretation is very frequentest in character, but Copenhagen is rather Bayesian.

Thanks
Bill

 

[TheAustrian]

I understand where your state of distraught comes from. In statistical mechanics we take the partition function of a system at equilibrium with an external heat bath and calculate quantities like average pressure, magnetic susceptibility, average energy etc. which all clearly have very important physical applications and are put to use extensively everyday in calculations.

In QM, if I have a pure state, what use are the expectation values of all the observables relevant to the system? What can I actually do with these expectation values? Sure I can calculate them at whim but can I put actually them to use in other calculations of physical applications? Do they hold the same all-important place that they do in statistical thermodynamics? You probably remember them showing up in non-degenerate and degenerate first order time-independent perturbation theory but where else do they show up?

Are these questions more representative of your own?

Why do I solve the Schrödinger Equation for? Why are Energy levels important? etc. I know nothing about the theory behind QM. I know only how to calculate questions that come up.

 

[atyy]

Why do I solve the Schrödinger Equation for? Why are Energy levels important? etc. I know nothing about the theory behind QM. I know only how to calculate questions that come up.

Schroedinger's equation is the notional counterpart to Newton's second law. It is the equation of motion. In classical mechanics, the state of a particle is specified by its position and momentum. If you know the state at one time, Newton's second law tells you the state any other time. In quantum mechanics, the state is a unit vector (more strictly speaking, the state is a ray that can be represented as a unit vector), such as the wave function. If you know the state at any time between two particular measurements, Schroedinger's equation tells you the state at any other time between those two particular measurements.

Energy levels are important for two reasons. The first reason is that energy levels can be measured by spectroscopy, so checking that quantum mechanics predicts the energy levels observed is an important check that quantum mechanics is correct.

A second, more technical reason that energy levels are important is that they determine the time evolution of a wave function that is an energy eigenstate. The time evolution of any wave function can be written as the superposition of the time evolution of individual energy eigenstates. But this technical reason is just applied mathematics, not physics.