## Thursday, July 29, 2010

### More Quantum Mechanics: Expectation Values

As I noted in a blog rebutting a fundie’s claim for a “precision universe”, that assumption fails because the cosmos at root is quantum mechanical in nature and not classical mechanical. This makes a world of difference, in particular, that we can’t hope to obtain exact or precise measurements but must use probabilities and uncertainties to get: “expectation values” (what we expect a position to be for example, in a given probability wave function configuration) or uncertainty computations.

In the case of expectation values, if the eigenfuction of an ensemble of particles is known (See Fig. 1), it may be used to calculate the average value that one would obtain when repeated measurements of that observable are made on the system. For any observable Q the expectation value, denoted as is given by the general formula (b) in Figure 1(which includes time,t - though sometimes dimension only is considered) . For the specific expectation value for a particle position in one dimension (say x- no dependence on time t) we need the form (c). Which we will soon compute.

Because the infinite square well (a) is bounded from 0 to L, we can replace the integral going from plus to minus infinity, with an easier to manage integral going from 0 to L.
This working to obtain the expectation value is given in Fig. 2 complete with the appropriate integrals, etc.

Note we take U = sin(kx) where k = pi/a, and U* = sin(kx) where k = pi/a (for n = 1 energy level) so:

UU* = sin^2(kx)

Quantum uncertainty and its computation, which we will get to in the next quantum instalment, measures the inherent spread in the results of measurements of an observable about the expectation value. The uncertainty in an observable (e.g. Q) with operator [Q] is given by the expression in Fig. 3. The key point is that if the uncertainty in the observable is zero, then the observable is said to be “sharp”. In other words, every measurement of Q made on the ensemble yields the same value. Thus there is no spread or distribution in the measurement.

Problem for next time: Consider the first excited state for an infinite square well, and obtain the expectation value in position (x) for it. Hint: the wave function U = sin (2pix/a).

Lastly, the energy level diagram for the solution to the previous problem is given in Fig. 4. The key step is to obtain the lowest energy state first, by getting: l’(max), s’(max) and j’(min). Thus: l’(max)= l1 + l2 = 0 + 1 = 1; s’(max) = ½ + ½ = 1 and j’(min) = [s’- l’] = 0. Then the lowest energy state is 3P_o. From LS coupling the other states of the multiplicity can easily be obtained (using s’ = 1, l’ =1)

Caleb Shay said...

I worked out the problem for next time, the expectation value for the n=2 excited state in the infinite square well and got:

= L/2

which turns out to be the same as the n=1 value. Is that right??

I also amazed myself by correctly solving the energy level diagram problem! I just followed all the steps you nicely developed from your previous QM blogs.

What you ought to make the fundies do is work these problems too! If they can't get them out they have no business arguing that quantum cosmologists are wrong and there was no spontaneous inception to the universe!

How about some more math blogs too?

mmfiore said...

As an alternative to Quantum Theory there is a new theory that describes and explains the mysteries of physical reality. While not disrespecting the value of Quantum Mechanics as a tool to explain the role of quanta in our universe. This theory states that there is also a classical explanation for the paradoxes such as EPR and the Wave-Particle Duality. The Theory is called the Theory of Super Relativity and is located at: Super Relativity
This theory is a philosophical attempt to reconnect the physical universe to realism and deterministic concepts. It explains the mysterious.

Copernicus said...

Caleb Shay wrote: How about some more math blogs too?
--

Check out the most recent one to do with Legendre polynomials! More will be coming.

Copernicus said...

mmfiore wrote:

"This theory is a philosophical attempt to reconnect the physical universe to realism and deterministic concepts."

Perhaps. But its doomed to being relegated to the dustbin of physics unless it achieves certain basic thresholds, none of which I have seen on the linked site.

For example, if it is to even be an adjunct to current QM what are its uncertainty relations (energy-time, position-momentum etc.).

Technically, the wave-particle duality is actually a reflection of the interfering observables in Quantum physics. These are usually expressed via the Poisson brackets (with non-commuting variables x, p):

[x, p] = -i h/ 2 pi

where h is the Planck constant of action.

If two variables a, b commute, then one has:

[a, b] = (a*b - b*a) = 0

if not, then:

[a,b] = (a*b - b*a) = -1

and we say a and b are 'non-commuting'.

What are the relevant ones for SR? If it is truly "deterministic" it will have two choices: either assert in all cased [p, q] = 0 (hardly plausible since all existing experiments refute it) OR [dp, dq] ~ h/2pi in which the form close the stochastic interpretation of Bohm-Hiley is used.

Next, it is disputable that the Michelson -Morley experiment can be overturned since its whole basis is predicated upon the *maximum* speed and Einstein defined this to be associated with light speed, not any lower "particle" speed. Indeed, using some lower velocity particle - whatever it may be - defeats the whole purpose of the experiment! (Which was to ascertain whether some *relative velocity* was feasible in conjunction with a velocity upper bound.

Next, to have the deterministic basis you claim - or SR claims -you'd have to have a physically REAL wave fucntion (PSI) as opposed to a statistical artifact. The only experiment I know which was designed to test this (the Gozzini -Rapisarda experiment) never was finished, and the result is therefore incomplete.

Finally, and most important, if this SR is a valid alternative than it ought to have papers published in a major peer-reviewed physics journal (like Physical Review D) but I've seen nothing. This shows me it has entertainment value only (like so many past "theories" I've seen that purport to overturn Einstein in some way) but no real validity.

Copernicus said...

mmfiore wrote:

"This theory is a philosophical attempt to reconnect the physical universe to realism and deterministic concepts."

Perhaps. But its doomed to being relegated to the dustbin of physics unless it achieves certain basic thresholds, none of which I have seen on the linked site.

For example, if it is to even be an adjunct to current QM what are its uncertainty relations (energy-time, position-momentum etc.).

Technically, the wave-particle duality is actually a reflection of the interfering observables in Quantum physics. These are usually expressed via the Poisson brackets (with non-commuting variables x, p):

[x, p] = -i h/ 2 pi

where h is the Planck constant of action.

If two variables a, b commute, then one has:

[a, b] = (a*b - b*a) = 0

if not, then:

[a,b] = (a*b - b*a) = -1

and we say a and b are 'non-commuting'.

What are the relevant ones for SR? If it is truly "deterministic" it will have two choices: either assert in all cased [p, q] = 0 (hardly plausible since all existing experiments refute it) OR [dp, dq] ~ h/2pi in which the form close the stochastic interpretation of Bohm-Hiley is used.

Next, it is disputable that the Michelson -Morley experiment can be overturned since its whole basis is predicated upon the *maximum* speed and Einstein defined this to be associated with light speed, not any lower "particle" speed. Indeed, using some lower velocity particle - whatever it may be - defeats the whole purpose of the experiment! (Which was to ascertain whether some *relative velocity* was feasible in conjunction with a velocity upper bound.

Next, to have the deterministic basis you claim - or SR claims -you'd have to have a physically REAL wave fucntion (PSI) as opposed to a statistical artifact. The only experiment I know which was designed to test this (the Gozzini -Rapisarda experiment) never was finished, and the result is therefore incomplete.

Finally, and most important, if this SR is a valid alternative than it ought to have papers published in a major peer-reviewed physics journal (like Physical Review D) but I've seen nothing. This shows me it has entertainment value only (like so many past "theories" I've seen that purport to overturn Einstein in some way) but no real validity.

Copernicus said...

mmfiore wrote:

"This theory is a philosophical attempt to reconnect the physical universe to realism and deterministic concepts."

Perhaps. But its doomed to being relegated to the dustbin of physics unless it achieves certain basic thresholds, none of which I have seen on the linked site.

For example, if it is to even be an adjunct to current QM what are its uncertainty relations (energy-time, position-momentum etc.).

Technically, the wave-particle duality is actually a reflection of the interfering observables in Quantum physics. These are usually expressed via the Poisson brackets (with non-commuting variables x, p):

[x, p] = -i h/ 2 pi

where h is the Planck constant of action.

If two variables a, b commute, then one has:

[a, b] = (a*b - b*a) = 0

if not, then:

[a,b] = (a*b - b*a) = -1

and we say a and b are 'non-commuting'.

What are the relevant ones for SR? If it is truly "deterministic" it will have two choices: either assert in all cased [p, q] = 0 (hardly plausible since all existing experiments refute it) OR [dp, dq] ~ h/2pi in which the form close the stochastic interpretation of Bohm-Hiley is used.

Next, it is disputable that the Michelson -Morley experiment can be overturned since its whole basis is predicated upon the *maximum* speed and Einstein defined this to be associated with light speed, not any lower "particle" speed. Indeed, using some lower velocity particle - whatever it may be - defeats the whole purpose of the experiment! (Which was to ascertain whether some *relative velocity* was feasible in conjunction with a velocity upper bound.

Next, to have the deterministic basis you claim - or SR claims -you'd have to have a physically REAL wave fucntion (PSI) as opposed to a statistical artifact. The only experiment I know which was designed to test this (the Gozzini -Rapisarda experiment) never was finished, and the result is therefore incomplete.

Finally, and most important, if this SR is a valid alternative than it ought to have papers published in a major peer-reviewed physics journal (like Physical Review D) but I've seen nothing. This shows me it has entertainment value only (like so many past "theories" I've seen that purport to overturn Einstein in some way) but no real validity.

janidebar said...

Good points, copernicus! At most SR is a conjecture and not even a very good one. To be a theory it has to at least have made its own discrete predictions, then confirmed them in actual observations. This SR has done neither.

That may also be why it's not been published in any real physics journals of the type you mentioned. How could it?