Integrating Gaussian Distribution (QM)

[Jimmy87]

Homework Statement

I am struggling with one of the end of chapter questions in my QM textbook (see attachment as I don't know how to show calculus on PF). It has thrown me because the chapter introduces some of the key principles in QM by talking about probability but then it randomly chucks in a question about the Gaussian distribution which is not covered at all in that chapter (or the whole book). Why have they put this question in there? What is the importance of this distribution to QM?
It asks you to solve for <x>. In the chapter it only briefly refers to <x> where it calls it an average and gives an example using some data. But what does it mean to take <x> (an average) of the Gaussian distribution?
I understand most of the integration but not the last bit. I don't get how there is a root pi over lambda in the final line at the end? If you look it up in the integral table it gives root pi over 2 lambda? Also in the table it says erf (u root lambda). Why does the solution not show this or do you ignore it? Also this integration in an integration of limits with respect to 'u' but how do you take the limits when the solution to the integral has no 'u' term in it?

Homework Equations

see attachment

The Attempt at a Solution

see attachment

 

[blue_leaf77]

I don't know how to show calculus on PF

Use latex to type maths https://www.physicsforums.com/help/latexhelp/.

Why have they put this question in there? What is the importance of this distribution to QM?

Gaussian distribution is frequently encountered as an example in the introductory chapters in QM book because its integration over all space can be easily computed analytically.

It asks you to solve for <x>. In the chapter it only briefly refers to <x> where it calls it an average and gives an example using some data. But what does it mean to take <x> (an average) of the Gaussian distribution?

In QM, wavefunctions of a system is interpreted to be the probability to find that system in a particular state. When the wavefunction is written as a function of spatial coordinate like it is in your book, it means that this wavefunction gives the probability to find the system at various position $x$. Calculating $\langle x \rangle$ for any form of probability distributions (or wavefunctions) has physical meaning of finding the average of the result of position measurement.

 

[DrClaude]

Learning to do Gaussian integrals is very important in physics. You can learn more here http://mathworld.wolfram.com/GaussianIntegral.html

It asks you to solve for <x>. In the chapter it only briefly refers to <x> where it calls it an average and gives an example using some data. But what does it mean to take <x> (an average) of the Gaussian distribution?

It means that if you pick a value randomly according to that distribution, what value of x will get on average.

I don't get how there is a root pi over lambda in the final line at the end?

See the link above.

If you look it up in the integral table it gives root pi over 2 lambda? Also in the table it says erf (u root lambda).

Are you sure this is for the same integration limits?

Also in the table it says erf (u root lambda). Why does the solution not show this or do you ignore it? Also this integration in an integration of limits with respect to 'u' but how do you take the limits when the solution to the integral has no 'u' term in it?

I am not sure what you mean here. They make the substitution u = x-a.

Homework Equations

see attachment

The Attempt at a Solution

see attachment

Next, please do not put things in a Word document. It is too much hassle to work with.

 

[Jimmy87]

Use latex to type maths https://www.physicsforums.com/help/latexhelp/.

Gaussian distribution is frequently encountered as an example in the introductory chapters in QM book because its integration over all space can be easily computed analytically.

In QM, wavefunctions of a system is interpreted to be the probability to find that system in a particular state. When the wavefunction is written as a function of spatial coordinate like it is in your book, it means that this wavefunction gives the probability to find the system at various position $x$. Calculating $\langle x \rangle$ for any form of probability distributions (or wavefunctions) has physical meaning of finding the average of the result of position measurement.

Thanks for your help. I also seen my error in my calculation. I have found a different website where the integral is indeed equal to root pi of lambda. I guess you don't use the common one from the integral table. So if you take <x> of a wave-function you are finding the average result of its position? Is this done very often? I thought the most important parameter when you are concerned with x is to find the probability which is the square of the wavefunction.

 

[Jimmy87]

Learning to do Gaussian integrals is very important in physics. You can learn more here http://mathworld.wolfram.com/GaussianIntegral.htmlIt means that if you pick a value randomly according to that distribution, what value of x will get on average.See the link above.Are you sure this is for the same integration limits?I am not sure what you mean here. They make the substitution u = x-a.Next, please do not put things in a Word document. It is too much hassle to work with.

Thanks for the help. It also asks you co calculate <x^2> (average of the squares) and sigma squared (variance) for the same Gaussian distribution which I am fine with for the integration but how are these values useful and what do they mean in the context of QM?

 

[DrClaude]

Expectation values of observables tell you what value will be measured, on average, for an ensemble of identically prepared quantum systems. Calculating $\langle x^2 \rangle$ by itself can be useful to find the average potential energy of a quantum harmonic oscillator. And doesn't $\sigma_x = \sqrt{\langle x^2 \rangle - \langle x \rangle^2}$ remind you of the Heisenberg uncertainty principle?

 

[blue_leaf77]

Thanks for the help. It also asks you co calculate <x^2> (average of the squares) and sigma squared (variance) for the same Gaussian distribution which I am fine with for the integration but how are these values useful and what do they mean in the context of QM?

I suggest that you read section 3.1 and 3.2 in this paper http://www.psiquadrat.de/downloads/ballentine70.pdf to get some preliminary insight about the relation between QM and statistics. As you seem to be new in QM, I encourage that you do not read the rest of the paper, just limit to those two sections for now.

 

[Jimmy87]

Expectation values of observables tell you what value will be measured, on average, for an ensemble of identically prepared quantum systems. Calculating $\langle x^2 \rangle$ by itself can be useful to find the average potential energy of a quantum harmonic oscillator. And doesn't $\sigma_x = \sqrt{\langle x^2 \rangle - \langle x \rangle^2}$ remind you of the Heisenberg uncertainty principle?

Could somebody tell me how people derive wavefunctions. In a textbook you are always given some wavefunction but how do you work out what the wavefunction should be? Suppose you want the wavefunction of an electron in the ground state of a hydrogen atom, how do you work out what the wavefunction should be? Or does that require one to be an expert at QM? Also, could someone tell me why wavefunctions are exponential? And why some are complex yet others are not?

I can solve the maths and I am progressing and solving the problems given to me but I have no idea of the implications behind what I am doing but I am not taught this so does one simply mindlessly follow the mathematics (because that is what I am doing)? For example, when I learned about capacitors and exponential decays I understood every variable in the equation and what it meant and where it comes from. Is it just too complicated to do this in QM?

 

[DrClaude]

There are actually very few cases where you can get analytical solutions to the Schrödinger equation. The important ones are particle in a box, plane waves, harmonic oscillator, and hydrogen atom (radial and angular parts). I suggest you take a good QM book and go through the derivations for those cases, especially the last two. That should help you understand why the eigenfunctions look the way they do.

For instance, the reason you get exponentials is that you need a functional form that, when derived twice, will give you back the same function. In addition, the wave function is required to decay to zero so that it is normalizable.

Wave functions are defined up to a complex phase. For most Hamiltonians, it turns out that you can always chose the complex phase such that the eigenfunctions are real. But the actual state of a quantum system will often require complex wave functions.

I don't think that QM is that much more complicated, at the level you are currently learning it, that you can't get the same understanding as you do for circuits. Sometimes is simply a question of finding the right material that will teach it in a way that makes sense to you.

 

[Jimmy87]

There are actually very few cases where you can get analytical solutions to the Schrödinger equation. The important ones are particle in a box, plane waves, harmonic oscillator, and hydrogen atom (radial and angular parts). I suggest you take a good QM book and go through the derivations for those cases, especially the last two. That should help you understand why the eigenfunctions look the way they do.

For instance, the reason you get exponentials is that you need a functional form that, when derived twice, will give you back the same function. In addition, the wave function is required to decay to zero so that it is normalizable.

Wave functions are defined up to a complex phase. For most Hamiltonians, it turns out that you can always chose the complex phase such that the eigenfunctions are real. But the actual state of a quantum system will often require complex wave functions.

I don't think that QM is that much more complicated, at the level you are currently learning it, that you can't get the same understanding as you do for circuits. Sometimes is simply a question of finding the right material that will teach it in a way that makes sense to you.

Great thanks for that information. That is a relief that these functions can be understood in depth. Do you recommend any books what will explain these as you said? I have Griffiths Intro to QM but there isn't much explaining in that sense.

 

[DrClaude]

The most mathematical introductory QM book I know is Walter Greiner's Quantum Mechanics: An Introduction. I think you will find the answers to your questions in that book.