# Energy levels

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Can a particle have an irrational amount of energy?

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Can a particle have an irrational amount of energy?

The limit of E is the total energy of the universe (if such a thing exists) so an irrational amount of energy would be more energy that exists. There's always the scenario where a tardyon (which always moves at the speed of light) moves faster than light. The energy of the particall would be conceptually irrational as well as mathematically a complex number, both of which are nonsense and therefore impossible.

Edited by pmb
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Can a particle have an irrational amount of energy?

If you mean by the mathematical definition of irrational (not a ratio of integers), then the question is to some extent pointless. The energy is defined in terms of the units that it is measured in joules, electron-volts, etc. The number of units is measured to a certain precision, i.e. within an error interval. In that interval there are an infinite number of rational numbers and an infinite number of irrational numbers, any of which can be the actual energy.

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It's moot, because you can't determine the exact energy, from $\Delta{E}\Delta{t}>\hbar$

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uncertainty in energy x uncertainty in time? > reduced planck's constant?

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You can calculate the energies concerned (to arbitrary accuracy) and those calculations are often in the forms of sums of infinite series so the answers are quite likely to be irrational.

Also, the length of a bit of string is (almost certainly) irrational.

Say it's about 15 cm long.

If I measure it with a ruler calibrated in cm I get an answer of 15 cm.

If I use a finer ruler I might get 15.3cm.

with a better measurement I get- say- 15.31. Each time I measure it more precisely, I get more places of decimals in the answer.

Even allowing for the nature of atoms, I can keep on getting more and more places of decimals and so (at least down to the Plank length) the measurement isn't rational.

I'm not certain what's meant to happen on a smaller scale than that but I suspect that you sometimes get 15. ...........01 cm and sometimes 15. ...........02 cm.

The more often you repeat the measurement, the closer you get to the true value, but the series never stops getting longer.

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uncertainty in energy x uncertainty in time? > reduced planck's constant?

Yes, the Heisenberg uncertainty principle. To get a precise measurement of energy takes a long time.

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thanks swansont,

i understand the heisenberg uncertainty principle, like delta momentum times delta velocity > reduced plancks constant over 2(from what iv heard)(please correct me if this is wrong)

i think there are lots of these relationships, how do i find them or work them out?

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thanks swansont,

i understand the heisenberg uncertainty principle, like delta momentum times delta velocity > reduced plancks constant over 2(from what iv heard)(please correct me if this is wrong)

i think there are lots of these relationships, how do i find them or work them out?

(yes, I was ignoring a factor of 2)

Others are from conjugate variables in QM; operators that don't commute are going to give you an uncertainty if you try and measure both. The other common one would be any two components of angular momentum.

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It's moot, because you can't determine the exact energy, from $\Delta{E}\Delta{t}>\hbar$

Search the concept "energy eigenfunction" in some textbook.

Yes, the Heisenberg uncertainty principle. To get a precise measurement of energy takes a long time.

The Heisenberg uncertainty relation is for non-commuting operators. Time is not an operator.

As Lev Landau once joked "To violate the time-energy uncertainty relation all I have to do is measure the energy very precisely and then look at my watch!" [22]

One false formulation of the energy-time uncertainty principle says that measuring the energy of a quantum system to an accuracy $\Delta E$ requires a time interval $\Delta t > h/\Delta E$. This formulation is similar to the one alluded to in Landau's joke, and was explicitly invalidated by Y. Aharonov and D. Bohm in 1961 [24].

Edited by juanrga
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Search the concept "energy eigenfunction" in some textbook.

That might be useful if I was unfamiliar with what an energy eigenfunction is.

The Heisenberg uncertainty relation is for non-commuting operators. Time is not an operator.

I noted the relation for non-commuting operators. Nevertheless, it also holds for energy and time. Which is explained in the wikipedia material you selectively quoted:

A state that only exists for a short time cannot have a definite energy. To have a definite energy, the frequency of the state must accurately be defined, and this requires the state to hang around for many cycles, the reciprocal of the required accuracy.

For example, in spectroscopy, excited states have a finite lifetime. By the time-energy uncertainty principle, they do not have a definite energy, and each time they decay the energy they release is slightly different. The average energy of the outgoing photon has a peak at the theoretical energy of the state, but the distribution has a finite width called the natural linewidth. Fast-decaying states have a broad linewidth, while slow decaying states have a narrow linewidth.

BTW, excited states are energy eigenfunctions.

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That might be useful if I was unfamiliar with what an energy eigenfunction is.

You would be also familiarized with the associated eigenvalues and their measurement.

I noted the relation for non-commuting operators.

Yes, but the point was that time is not an operator and, thus, expression $\Delta{E}\Delta{t}>\hbar$ is not a Heisenberg uncertainty relation, but something with a different physical meaning.

Nevertheless, it also holds for energy and time. Which is explained in the wikipedia material you selectively quoted:

I quoted the relevant parts explaining why what you said about the measurement of energy and the time of measurement was false.

I can cite other parts. For instance, the phrase just after that last that I cited above explains that is not the measurement time for excited states, as it seems you still believe:

The time in the uncertainty relation is the time during which the system exists unperturbed, not the time during which the experimental equipment is turned on
Edited by juanrga
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I can cite other parts. For instance, the phrase just after that last that I cited above explains that is not the measurement time for excited states, as it seems you still believe:

Because I quoted the section on excited states? Narrow states are long-lived.

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It's moot, because you can't determine the exact energy, from $\Delta{E}\Delta{t}>\hbar$

That relation is not a strict Heisenberg relationship. It doesn't mean that energy can't be measured precisely.

Recall the derivation of that expression and the meaning of E in it.

http://home.comcast.net/~peter.m.brown/qm/time_energy_hup.htm

uncertainty in energy x uncertainty in time? > reduced planck's constant?

There is no such thing as uncertainty in time. A real Heisenberg expression requires two observables, Time is not an observable and as such there is no time operator. Time is a paramenter.

That relation is not a strict Heisenberg relationship. It doesn't mean that energy can't be measured precisely.

Recall the derivation of that expression and the meaning of E in it.

http://home.comcast.net/~peter.m.brown/qm/time_energy_hup.htm

There is no such thing as uncertainty in time. A real Heisenberg expression requires two observables, Time is not an observable and as such there is no time operator. Time is a paramenter.

juanrga - Excellant!! Way to go! You"ve got it precisely right in my opinion.

Some people were making those kinds of arguements to me a long time ago. I found that a lot of people misuse that relationship that swansont. I also found that those people didn't understand time. They thought it was an observable.

Griffiths has a nice insight into this. From Introduction to Elementary Particlesby DAvid Griffiths, 2004). See page 51-52, regarding that relation

Nevertheless, it is a useful device for "back-of-the-envelope" calculations, and it does very well for the pi meson. Unfortunately, many books present it as though it were a rigorous derivation, which it certainly is not. The uncertainly principle does not license violation of conservation of energy (nor does any such violation occur in this process.... In general, when you hear a physicist invoke the uncertainty principle, keep your hand on your wallet.

Oh how I love that quote!

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That relation is not a strict Heisenberg relationship.

Fair enough: it's an uncertainty relationship but not Heisenberg.

It doesn't mean that energy can't be measured precisely.

But that wasn't the question being addressed. The question is more along the lines of must energy be able to be measured precisely, because the OP wanted to know if something can have an irrational amount of energy. I was simply pointing out that there are states where the energy is not precisely defined.

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juanrga - Excellant!! Way to go! You"ve got it precisely right in my opinion.

Some people were making those kinds of arguements to me a long time ago. I found that a lot of people misuse that relationship that swansont. I also found that those people didn't understand time. They thought it was an observable.

Yes. Thank you.

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Fair enough: it's an uncertainty relationship but not Heisenberg.

But that wasn't the question being addressed. The question is more along the lines of must energy be able to be measured precisely, because the OP wanted to know if something can have an irrational amount of energy. I was simply pointing out that there are states where the energy is not precisely defined.

Sorry. My mistake. I also mistakenlky said that you used it incorrectly and I'm not in a position to say such a thing. Sorry Tom.

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So can a particle have exactly root 2 energy?

What is the uncertainty between time and energy then? Is the uncertainty in time do do with uncertainty in velocity and special/general relativity?

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There's nothing inherently unphysical in having root 2 for an energy.

The time uncertainty is e.g. the lifetime of an excited state; a state that has a finite lifetime cannot have an exact energy. Conceptually you can think of a pure tone — a single frequency. It must be of infinite extent in space, and therefore in time, in order to be a pure tone. If you make it finite, you add in Fourier components, i.e. frequencies that are not at the main value, so there is now an uncertainty in the frequency.

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What do you mean there is nothing inherently unphysical about having root 2 energy?

Ok, so say an electron is in an excited state, do we know how much energy it has quite precisely, so we can't say how long it will be in that state for very precisely?

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What do you mean there is nothing inherently unphysical about having root 2 energy?

Ok, so say an electron is in an excited state, do we know how much energy it has quite precisely, so we can't say how long it will be in that state for very precisely?

In the excited state you cannot precisely say what energy it has. You would not be able to preclude some irrational value.

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