Why do uncertainty relations not apply in stationary states?

In summary, the conversation discusses the nature of uncertainty relations in quantum mechanics, specifically the relation between energy and position uncertainty. The book referenced suggests that this relationship has no meaning in a stationary state, which is a wave function with a constant probability through time. The link provided offers further explanation on this topic. The conversation also touches on the difference between the uncertainty relationship between position and momentum and the one between energy and time, with the latter being a property of a probability distribution rather than a measurable observable. The conversation concludes with a clarification on the definition of a stationary state and its properties.
  • #1
RPI_Quantum
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I am trying to understand the nature of uncertainty relations in quantum mechanics. I am looking specifically at a relation between energy and position uncertainty... the book that I am reading hints that this relationship has no meaning in a stationary state. Why would that be?
 
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  • #2
Maybe if I were to say that a Sationary state (eigenstate) is a wave function who's probablity is constant through time? Take a look thorugh here: http://www.chemistry.ohio-state.edu/betha/qm/ [Broken]
 
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  • #3
Thanks

Yes that did help a lot. However, I think I am still missing something. I understand that in the stationary state nothing changes with time. Yet, I do not quite see how this makes the uncertainty relationship between energy and position any different from the one between position and momentum. Maybe if you could elaborate on how the constant probability effects this uncertainty relation...

Thanks for the link by the way. I didn't search through it entirely yet, but many questions that I have been having were certainly addressed in it.
 
  • #4
RPI_Quantum said:
Yes that did help a lot. However, I think I am still missing something. I understand that in the stationary state nothing changes with time. Yet, I do not quite see how this makes the uncertainty relationship between energy and position any different from the one between position and momentum.

In non-relativistic QM, there is difference in nature between the x-p uncertainty and the E-t uncertainty.
x and p are observables, so they both are results of measurements in NR QM ; the uncertainty principle just gives you a property of their probability distributions (namely that the product of their standard variations has a lower bound).
E is also an observable, but time (t) is a parameter. There is NO T - operator or T observable in QM. So you cannot talk about "an uncertainty in time" as the standard deviation of the probability distribution of t.

But what then does the E - t "uncertainty" relation mean ?

It actually means that if you want to have significant changes in the expectation values of ANY observable within a time lapse dt, then you need to have a superposition of several energy eigenstates such that the standard deviation of the energy measurement becomes at least dE, with
dt . dE > hbar/2

cheers,
Patrick.
 
  • #5
I would just like to add that your posts are very clear and helpful, Patrick!
 
  • #6
RPI_Quantum,

Does your "RPI" stand for "Rennselaer Polytechnic Institute"?

I went to RPI, and I now teach at HVCC. Hi! :smile:
 
  • #7
DB said:
Maybe if I were to say that a Sationary state (eigenstate) is a wave function who's probablity is constant through time? Take a look thorugh here: http://www.chemistry.ohio-state.edu/betha/qm/ [Broken]

That's an inexact formulation of a property of the stationary states,not exactly the DEFINITION.Stationary states in the standard formulation are described through NORMALIZED EIGENSTATES OF THE TIME-INDEPENDENT HAMILTONIAN (Schroedinger picture) ASSOCIATED TO EIGENVALUES (denoted generally by [itex]E_{n} \in \sigma_{d}(\hat{H}) [/itex])...

What wave function and what probability are u talking about...??


Daniel.
 
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  • #8
Indeed the post of Vanesch explains everything and it is very well written. Now if you just realized that a stationary state is represented by a wavefunction with only ONE energy-value in it (not a superposition of several energy values [tex]E_{i}[/tex]), it is quite straightforward to see that the uncertainty-relations do not apply here. Just focus on the example that Vanesch gave as to explain the uncertainty between both E and t

regards
marlon

ps : indeed, the probability density of a stationary state is independent of time but NOT position and the expectation value of any observable (provided the observable is independent of t itself) is also independent of time BUT NOT position.
 
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1. What are stationary states?

Stationary states, also known as energy eigenstates, are quantum states in which the probability of finding a particle does not change over time. This means that the energy of the particle remains constant and does not change with time.

2. How are stationary states different from non-stationary states?

Non-stationary states, also known as energy superpositions, are quantum states in which the probability of finding a particle changes over time. This means that the energy of the particle is not constant and can change with time.

3. What is the significance of stationary states?

Stationary states play a crucial role in quantum mechanics as they represent the only states in which the energy of a particle is well-defined. They also serve as the basis for understanding the behavior of quantum systems and predicting their future behavior.

4. How are stationary states calculated or determined?

Stationary states can be calculated using the Schrödinger equation, which is a fundamental equation in quantum mechanics. This equation describes the time evolution of quantum systems and allows for the determination of stationary states and their corresponding energies.

5. Are stationary states always observed in quantum systems?

No, stationary states are not always observed in quantum systems. This is because quantum systems can exist in both stationary and non-stationary states, and their state can change depending on external factors such as interactions with other particles or measurements.

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