Energy*time uncertainty for particle decay

[jjson775]

TL;DR

If a particle decays into two gamma rays, how do you calculate the energy time uncertainty?

If a particle decays into two gamma rays, what is the energy time uncertainty?
ΔEΔt = h bar or
ΔEΔt = h bar/2
Please explain.

 

[PeterDonis]

what is the energy time uncertainty?

There is no such thing. Energy and time are not complementary observables, so there is no uncertainty relation between them. Indeed, time isn't an observable at all, it's a parameter.

It is possible, if you pick some observable other than energy, to construct a relation that sort of kind of looks like an uncertainty relation between energy (the Hamiltonian) and the time it takes for that observable to change significantly. But I don't think that's what you're asking about.

More here:

https://math.ucr.edu/home/baez/uncertainty.html

 

[jjson775]

There is no such thing. Energy and time are not complementary observables, so there is no uncertainty relation between them. Indeed, time isn't an observable at all, it's a parameter.

It is possible, if you pick some observable other than energy, to construct a relation that sort of kind of looks like an uncertainty relation between energy (the Hamiltonian) and the time it takes for that observable to change significantly. But I don't think that's what you're asking about.

More here:

https://math.ucr.edu/home/baez/uncertainty.html

Sorry for the sloppy post. Nevertheless, I am confused by your reply. I am a retired engineer, 83 years old, self studying modern physics. All of the textbooks I use show the uncertainty principle for the energy and time interval:
ΔEΔt ≥ h bar/2. My question is how to calculate the fractional uncertainty in the mass determination, Δm/m, when the particle decays in a given time interval into two gamma rays. To find ΔE, do I use h bar/2 or just h bar? Why?

 

[PeroK]

Sorry for the sloppy post. Nevertheless, I am confused by your reply. I am a retired engineer, 83 years old, self studying modern physics. All of the textbooks I use show the uncertainty principle for the energy and time interval:
ΔEΔt ≥ h bar/2.

Introduction to QM by Griffiths explodes that myth. There are a lot of sloppy textbooks it appears.

 

[PeroK]

PS my Internet is down. I think I posted a quotation from Griffiths on this recently if you can find it.

 

[jjson775]

Introduction to QM by Griffiths explodes that myth. There are a lot of sloppy textbooks it appears.

What?? I am shocked. Among others, I went through the modern physics chapters of Young and Freedman, 14th edition 2016, supposedly the successor to Sears and Zemansky when I studied physics (classical) over 60 years ago. Have they dropped this (E,t uncertainty)? How about position and momentum?

 

[JimWhoKnew]

PS my Internet is down. I think I posted a quotation from Griffiths on this recently if you can find it.

Did you mean:

In general, when you hear a physicist invoke the uncertainty principle, keep a hand on your wallet.

?

(my favorite Griffiths quotation)

 

[PeterDonis]

Among others, I went through the modern physics chapters of Young and Freedman, 14th edition 2016, supposedly the successor to Sears and Zemansky when I studied physics (classical) over 60 years ago.

I note you say (classical). Those textbooks are not focused on QM; they might cover it, but they're not primarily QM textbooks. Which means, unfortunately, that they will "teach" you things you'll have to unlearn if you actually want to study QM.

Have they dropped this (E,t uncertainty)?

No, it was never a valid concept in the first place. A textbook that is focused on QM, like Griffiths, will tell you that.

How about position and momentum?

Position and momentum are both observables, and are complementary (non-commuting observables over the same Hilbert space), so there is a valid uncertainty relation between them, the one you're used to.

A proper textbook will not just state "uncertainty relation" by fiat, but will explain to you why they exist. I just gave a brief explanation of why the position-momentum uncertainty relation exists--and earlier I explained why an energy-time uncertainty relation doesn't exist. Do those explanations make sense? If not, please ask questions so you can better understand.

 

[WernerQH]

If a particle decays into two gamma rays, what is the energy time uncertainty?
ΔEΔt = h bar or
ΔEΔt = h bar/2
Please explain.

I think the first alternative is better, but it depends on how you define ΔE and Δt. For $\Delta t$ it is customary to use the lifetime of the decaying particle, and for $\Delta E$ the full-width-half-maximum (FWHM) of the line in the energy spectrum. The line shape is typically fitted with a Lorentzian, but the variance of such a Cauchy distribution is infinite! That's why one resorts to using the FWHM measure.

The line shape can be written in the form $$
{1 \over \pi} {{\Gamma / 2} \over {(\omega - \omega_0)^2 + (\Gamma / 2)^2}}
$$ and from this you find for $\omega - \omega_0 = \pm \Gamma / 2$: $$
(\Delta \omega)_\text{FWHM} = \Gamma = 1 / \tau
$$ with $\tau$ the lifetime of the excited state. Dividing by $2 \pi$ to convert from angular to normal frequencies, and multiplying by $h$ I find $$
\Delta E = {h \over {2 \pi \tau}} = {\hbar \over \tau} \ .
$$

You see, this is only indirectly related to the "usual" uncertainty relations.

 

[Herman Trivilino]

Have they dropped this (E,t uncertainty)? How about position and momentum?

The position-momentum uncertainty principle is alive and well.

I don't know the history, but certainly one still sees the erroneous energy-time uncertainty relation in publications, and prominent physicists still use it in their explanations.

This might be like the way the uncertainty principle itself is discussed. When Heisenberg first came up with it he explained it as a measurement issue, often called the disturbance principle. For example, if you know the momentum of a particle with little uncertainty and try to measure its position you disturb it, destroying very thing you're trying to measure. But a short time later he realized it's not that you're disturbing a well-defined property (like position), it's that the property itself is not well-defined.

So, for example, trying to determine the position of an atomic electron is a problem, not because the act of measuring it disturbs its position, but because the property of position is not a well-defined property of an atomic electron.

Yet we still see in publications and hear explanations from prominent physicists the uncertainty principle being described as the disturbance principle.

 

[jjson775]

I note you say (classical). Those textbooks are not focused on QM; they might cover it, but they're not primarily QM textbooks. Which means, unfortunately, that they will "teach" you things you'll have to unlearn if you actually want to study QM.


No, it was never a valid concept in the first place. A textbook that is focused on QM, like Griffiths, will tell you that.


Position and momentum are both observables, and are complementary (non-commuting observables over the same Hilbert space), so there is a valid uncertainty relation between them, the one you're used to.

A proper textbook will not just state "uncertainty relation" by fiat, but will explain to you why they exist. I just gave a brief explanation of why the position-momentum uncertainty relation exists--and earlier I explained why an energy-time uncertainty relation doesn't exist. Do those explanations make sense? If not, please ask questions so you can better understand.

To clarify, I studied “classical” physics 60+ years ago as an engineering student. In recent years, I have studied modern physics, relativity, Planck, the basics of QM, etc.

 

[PeroK]

This might be like the way the uncertainty principle itself is discussed. When Heisenberg first came up with it he explained it as a measurement issue, often called the disturbance principle. For example, if you know the momentum of a particle with little uncertainty and try to measure its position you disturb it, destroying very thing you're trying to measure. But a short time later he realized it's not that you're disturbing a well-defined property (like position), it's that the property itself is not well-defined.

I found a video of an Indian physics professor exploding this myth for his students. He knew they would all have learned the "old" (disturbance) UP and was clearly practised at retraining his students to understand the real thing.

 

[PeroK]

To clarify, I studied “classical” physics 60+ years ago as an engineering student. In recent years, I have studied modern physics, relativity, Planck, the basics of QM, etc.

A lot of what passes for QM in non-specialist courses is a messy hybrid of QM and classical thinking, IMHO. If I could explain why professors and authors choose to teach that hybrid stuff I would, but I don't know. It's a bit like playing Bohemian Rhapsody and telling you that's grand opera!

The real stuff is much more interesting.

Also, the ET UP makes zero sense to me the way it's taught. Measure the energy of a system now and in a million years (large $\Delta t$) and the difference in energy measurements ($\Delta E$) must be small? That can't be right!

 

[PeterDonis]

the basics of QM

But have you studied QM from an actual QM textbook? It doesn't seem like it; it seems like you've been trying to learn it from the same kind of general textbooks you used to learn "classical" physics. It doesn't seem like that approach works well.

 

[Herman Trivilino]

I found a video of an Indian physics professor exploding this myth for his students. He knew they would all have learned the "old" (disturbance) UP and was clearly practised at retraining his students to understand the real thing.

I would love to see that if you can dig it up.

 

[jjson775]

But have you studied QM from an actual QM textbook? It doesn't seem like it; it seems like you've been trying to learn it from the same kind of general textbooks you used to learn "classical" physics. It doesn't seem like that approach works well.

Thanks for the responses. In retrospect I should have posted this as introductory physics homework help. I am not at the level of the discussion in this thread. I love learning about modern physics but am limited to calculus and differential equations, no linear algebra or multi variable calculus so don’t plan to get into specialized QM.

 

[PeterDonis]

In retrospect I should have posted this as introductory physics homework help.

That wouldn't make the question answerable. Also, a homework question is best posed based on an actual homework question, i.e., something you found in an actual textbook, not something you just came up with.

For example, you could look in the introductory QM textbook by Griffiths that was referenced earlier. It has some actual homework problems on energy-time uncertainty after Section 3.4.3 (which is the section @PeroK was referring to earlier, that explains why there is no standard "uncertainty principle" between energy and time--the explanation given there is basically the same one that's been given in this thread).

Doing that might also help you to formulate a question along the lines of the one you asked in the OP of this thread, that actually would be answerable.

 

[PeterDonis]

am limited to calculus and differential equations, no linear algebra or multi variable calculus so don’t plan to get into specialized QM.

I'm not sure you can even do basic QM if you don't know at least some linear algebra.

 

[jjson775]

I think the first alternative is better, but it depends on how you define ΔE and Δt. For $\Delta t$ it is customary to use the lifetime of the decaying particle, and for $\Delta E$ the full-width-half-maximum (FWHM) of the line in the energy spectrum. The line shape is typically fitted with a Lorentzian, but the variance of such a Cauchy distribution is infinite! That's why one resorts to using the FWHM measure.

The line shape can be written in the form $$
{1 \over \pi} {{\Gamma / 2} \over {(\omega - \omega_0)^2 + (\Gamma / 2)^2}}
$$ and from this you find for $\omega - \omega_0 = \pm \Gamma / 2$: $$
(\Delta \omega)_\text{FWHM} = \Gamma = 1 / \tau
$$ with $\tau$ the lifetime of the excited state. Dividing by $2 \pi$ to convert from angular to normal frequencies, and multiplying by $h$ I find $$
\Delta E = {h \over {2 \pi \tau}} = {\hbar \over \tau} \ .
$$

You see, this is only indirectly related to the "usual" uncertainty relations.

Thanks. This answered my question.

 

[jjson775]

That wouldn't make the question answerable. Also, a homework question is best posed based on an actual homework question, i.e., something you found in an actual textbook, not something you just came up with.

For example, you could look in the introductory QM textbook by Griffiths that was referenced earlier. It has some actual homework problems on energy-time uncertainty after Section 3.4.3 (which is the section @PeroK was referring to earlier, that explains why there is no standard "uncertainty principle" between energy and time--the explanation given there is basically the same one that's been given in this thread).

Doing that might also help you to formulate a question along the lines of the one you asked in the OP of this thread, that actually would be answerable.

My question came from a homework problem from Modern Physics, Serway Chapter 5, Matter Waves. Yes, not a real QM textbook.

 

[jjson775]

I'm not sure you can even do basic QM if you don't know at least some linear algebra.

I don’t know what “basic QM” is but am happy to have gained some insight into quantized energy, wave particle duality, the Schrödinger equation etc. I am continually learning more all the time, eg., the shaky nature of energy time uncertainty!

 

[PeroK]

My question came from a homework problem from Modern Physics, Serway Chapter 5, Matter Waves. Yes, not a real QM textbook.

You have to go for it and learn and study matter waves if that's what the book does. Matter waves were part of an early version of QM that was overtaken by modern QM in the 1920s and 30s. Maybe I'm too skeptical and there is some benefit in learning these older formulations of QM: matter waves and the old UP.

 

[PeroK]

That said, I'd be wary of putting too much emphasis on either QM or SR, as presented in a general physics book. For me, both these subjects entailed such a fundamental departure from what I would have known as "classical physics", that I'm glad I learned from dedicated sources, where the subjects were given full reign.

 

[PeterDonis]

I don’t know what “basic QM” is

The kind that's presented in introductory QM textbooks, like Griffiths. In other words, nothing "specialized", just the basics of the QM framework and how it works.

 

[jjson775]

The kind that's presented in introductory QM textbooks, like Griffiths. In other words, nothing "specialized", just the basics of the QM framework and how it works.

I would appreciate an example of a topic in the “basics of the QM framework”.

 

[PeterDonis]

I would appreciate an example of the basics of the QM framework.

That's not a question that really makes sense. Suppose I asked you to give me an example of the basics of arithmetic. What would you say?

The way to find out "the basics of the QM framework" is to consult a textbook like the one that's been referenced repeatedly in this thread. You clearly have the time to read through textbooks; you just haven't chosen the right ones up to now to learn QM properly. The way to fix that is to change which textbooks you choose to spend your time reading to learn QM.

 

[jjson775]

You have to go for it and learn and study matter waves if that's what the book does. Matter waves were part of an early version of QM that was overtaken by modern QM in the 1920s and 30s. Maybe I'm too skeptical and there is some benefit in learning these older formulations of QM: matter waves and the old UP.

Is the DeBroglie wavelength still a valid concept?

 

[PeterDonis]

Is the DeBroglie wavelength still a valid concept?

Depends on what you mean. The de Broglie formula still appears in heuristic discussions of momentum in quantum mechanics (see, for example, Griffiths, Section 1.6), but the idea that the wavelength in that formula represents the actual wavelength of an actual wave is not valid. It's just a property of the quantum wave function.

Please note that it is not a good use of either your or others' time to keep asking questions like this. What you really are asking for amounts to being given a course in introductory QM. That's not what PF is for. You need to take the time to learn QM properly from a textbook like Griffiths. If, while doing that, you have questions that the textbook doesn't answer, you can start a new thread here to ask them.

In the meantime, this thread is closed since your question in the OP is answered.