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

Thanks

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- Thread starter forcefield
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- #1

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

Thanks

- #2

mfb

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- #3

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

- #4

mfb

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- #6

Simon Phoenix

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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.

- #7

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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.

- #8

blue_leaf77

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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?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.

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Simon Phoenix

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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

- #10

mfb

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Only phases in the basis where you want to calculate the probabilities, at the time when you want to calculate them.You know there is this so called phase that you can throw away when you calculate probabilities in QM.

No.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.

- #11

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

- #12

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

- #13

mfb

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blue_leaf77

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The only entity which connects QM with probabilistic interpretation is the wavefunction.No, I'm not considering wave functions. Actually I haven't studied them seriously yet.

- #15

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I think one should consider a period of π here.

- #16

mfb

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That does not make sense.I think one should consider a period of π here.

- #17

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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.That does not make sense.

- #18

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Sorry, but that made even less 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.

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