QM probability and normal distribution

[forcefield]

Is there a relationship between QM probability and normal distribution ?

I'm thinking about drawing probability densities as functions of phase.

Thanks

 

[mfb]

You can find things that follow a normal distribution in QM, but I would not call this "relationship between QM and normal distribution" in the same way I don't see a "relationship between the normal distribution and the number 3".

 

[forcefield]

You can find things that follow a normal distribution in QM, but I would not call this "relationship between QM and normal distribution" in the same way I don't see a "relationship between the normal distribution and the number 3".

You dropped out an essential word which was "probability". I'm looking for a mathematical relationship to calculate QM probabilities from probability densities.

 

[mfb]

Probability densitites are used in quantum mechanics, but quantum mechanics is more than pure probability theory.

 

[forcefield]

quantum mechanics is more than pure probability theory.

That's why I said QM probability.

 

[Simon Phoenix]

Hi Forcefield, I'm really not at all sure what you're trying to get at here.

The probability function you get depends on what observable you're trying to measure - and the state of the system. Suppose we prepared a single qubit (a spin-1/2 particle say) in an eigenstate of the spin-z operator, and let's suppose we prepared the spin up state. Then, assuming ideal measurements, the probability of getting spin up in a measurement of spin-z is 1, and the probability of getting spin down is 0. Not really very Gaussian :-)

Now suppose we measure spin-x, then the probability of getting a spin up result is now 1/2 and the probability of getting a spin down result is 1/2. A uniformly random distribution.

Let's take some more examples - if we have a coherent state of light and make a measurement of photon number - then we'll get a Poisson distribution (in many runs of the same experiments, of course). Measurement of the field quadrature operator of the same coherent state will give us a Gaussian (if I recall correctly). Take a photon number state of the EM field and measure it's phase - you'll get a uniformly random distribution.

So the probabilities depend crucially on what property we're choosing to measure and what state the system is in.

Could you perhaps be a bit more specific because your question doesn't make a lot of sense to me.

 

[forcefield]

Hi Simon,

Then, assuming ideal measurements, the probability of getting spin up in a measurement of spin-z is 1, and the probability of getting spin down is 0. Not really very Gaussian :-)

Actually I think it is Gaussian but the variance is zero.

Now suppose we measure spin-x, then the probability of getting a spin up result is now 1/2 and the probability of getting a spin down result is 1/2. A uniformly random distribution.

I think that makes it Gaussian too just like throwing coins. It is just difficult to see because there are only two options.

Let's take some more examples - if we have a coherent state of light and make a measurement of photon number - then we'll get a Poisson distribution (in many runs of the same experiments, of course). Measurement of the field quadrature operator of the same coherent state will give us a Gaussian (if I recall correctly). Take a photon number state of the EM field and measure it's phase - you'll get a uniformly random distribution.

I have only a vague idea what you are talking about here. Remember I marked this thread high school level.

Could you perhaps be a bit more specific because your question doesn't make a lot of sense to me.

You know there is this so called phase that you can throw away when you calculate probabilities in QM. I am thinking about whether it is possible to calculate QM probabilities by assuming that the measurement probability changes like normal distribution as time goes by.

 

[blue_leaf77]

I am thinking about whether it is possible to calculate QM probabilities by assuming that the measurement probability changes like normal distribution as time goes by.

Given $\psi(x,t)$, you want $|\psi(x,t)|^2$ at fixed position $x=x_0$ to follow Gaussian profile with time, is that what you mean?

 

[Simon Phoenix]

You know there is this so called phase that you can throw away when you calculate probabilities in QM

You're talking here about a global phase factor - you certainly can't ignore the relative phases between terms in a superposition when calculating probabilities. Recall that to calculate a probability in QM we're taking the square modulus of sums of complex numbers.

I think that makes it Gaussian too just like throwing coins

Well only in the sense that for sufficiently large numbers of trials then the Gaussian is a good approximation to the binomial (I think the rule of thumb is that np ≥ 5 where n is the number of trials and p is the probability)

 

[mfb]

You know there is this so called phase that you can throw away when you calculate probabilities in QM.

Only phases in the basis where you want to calculate the probabilities, at the time when you want to calculate them.

I am thinking about whether it is possible to calculate QM probabilities by assuming that the measurement probability changes like normal distribution as time goes by.

No.

 

[forcefield]

Given $\psi(x,t)$, you want $|\psi(x,t)|^2$ at fixed position $x=x_0$ to follow Gaussian profile with time, is that what you mean?

No, I'm not considering wave functions. Actually I haven't studied them seriously yet.

 

[forcefield]

You're talking here about a global phase factor - you certainly can't ignore the relative phases between terms in a superposition when calculating probabilities. Recall that to calculate a probability in QM we're taking the square modulus of sums of complex numbers.

Yes, I thought about that and it looks like a relative phase of pi/2 is required.

 

[mfb]

The relative phase in a superposition can be everything (well, as it is a phase, it is usually restricted to the range from 0 to 2pi).

 

[blue_leaf77]

No, I'm not considering wave functions. Actually I haven't studied them seriously yet.

The only entity which connects QM with probabilistic interpretation is the wavefunction.

 

[forcefield]

The relative phase in a superposition can be everything (well, as it is a phase, it is usually restricted to the range from 0 to 2pi).

I think one should consider a period of π here.

 

[mfb]

I think one should consider a period of π here.

That does not make sense.

 

[forcefield]

That does not make sense.

Yes, I was sloppy there. The length of the period doesn't really matter but if you look at the shape of normal distribution it is more like half-circle than full-circle.

 

[Heinera]

Yes, I was sloppy there. The length of the period doesn't really matter but if you look at the shape of normal distribution it is more like half-circle than full-circle.

Sorry, but that made even less sense.