Questions about expectation values and definite values (quantum physics)

In summary, the expectation value of momentum/position/energy is the average value we would expect to measure if we repeatedly measured particles in the same state. This does not necessarily mean that most of the particles will have the same momentum/position/energy, as it depends on the probability distribution generated from the wavefunction. The momentum of a particle described by a wavefunction can be measured through methods such as diffraction and scattering, but in practice, it is measured through tracking the particle's path. It is not possible for a wavefunction to have both definite momentum and position, as the uncertainty principle dictates a trade-off between the uncertainties in these two variables.
  • #1
kehler
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Is the expectation value of momentum/position/energy the value that we're most likely to measure? So suppose we measure 100 particles with the same wavefunction, would we expect most of them to have momentum/position/energy that's equal to the expectation value? And I was wondering, how do we even measure the momentum of a particle described by a wavefunction?

Also, suppose a wavefunction is found to have a definite value of momentum/position/energy, does this necessarily mean that the expectation value of its momentum/position/K.Energy will be equal to that definite value?
And lastly, is it possible to have a wavefunction that has both definite momentum and position?

I'm a bit confused by these concepts. Any help would be much appreciated :)
 
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  • #2
kehler said:
Is the expectation value of momentum/position/energy the value that we're most likely to measure?
no if you made the same measurement on repeatedly on particles perpeared in the same state it is the average of all you measurements. Consider a particle in a even superposition of spin up or down states (z axis). The expectation value is 0 - If you measure spin in the z axis, you will only ever find it up or down, 50% of the time for each in average, never zero.
kehler said:
So suppose we measure 100 particles with the same wavefunction, would we expect most of them to have momentum/position/energy that's equal to the expectation value?
no you would expect the average of all your measurements to be the expectation value. Whether it is most of them will depend on the probaility distribution gerated from the wavefunction
kehler said:
And I was wondering, how do we even measure the momentum of a particle described by a wavefunction?
not totally sure, there's probably many differnt methods & perhaps someone can correct me, but something like measuring the path in magnetic field(eg. bubble chamber) will give a charge to speed ratio and so yield momentum... (though that raises an interesting question as to measure path you are measureing position...)
kehler said:
Also, suppose a wavefunction is found to have a definite value of momentum/position/energy, does this necessarily mean that the expectation value of its momentum/position/K.Energy will be equal to that definite value?
theoretically yes though it would mean you other variable is totally unkown, which is not very physical, relations are
dp.dx and dE.dt >= hbar/2
kehler said:
And lastly, is it possible to have a wavefunction that has both definite momentum and position?
no, the unceratinty princlple give the product of the unceratinties as:
dp.dx >= hbar/2
so if there is only dx uncertainty in x, there must be at least dp = hbar/(2dx) uncertainty in p
kehler said:
I'm a bit confused by these concepts. Any help would be much appreciated :)
google could help as well there's quite a few discussions going round, or you could try the PF library...
 
  • #3
Thanks :). That helped.
 
  • #4
kehler said:
... how do we even measure the momentum of a particle described by a wavefunction?
If you really want it to be described by a wavefunction, then diffraction and scattering come to mind, such as sending the particle through a grating (or crystal lattice). deBroglie told us that the wavelength is inversely proportional to the momentum through the proportionality constant, h, so if you can determine the wavelength, then you can determine the momentum. Notice that diffraction and scattering obscure position, so there's your uncertainty principle at work.

The way momentum is measured in practice is basically what lanedance said. However, this relies on the particle nature rather than the wave nature, so it isn't really like measuring the momentum of the wavefunction. In the big colliders, they use silicon, wire mesh, and electronic channel readout instead of superheated liquid and photography, but the kinematical concept is the same: the momentum is proportional to the radius of curvature of the track.

I will also point out that the tracking measurement that lanedance and I described requires some finite length of track, which translates into an uncertainty in position (in the same direction as the momentum being measured, as it should). The longer the track the better the momentum measurement the more uncertain is the position. So there's your uncertainty principle at work again. You cannot escape from it even if the particle is acting like a particle. (Actually, that really isn't the uncertainty principle at work, it's detector resolution, which is much much worse than uncertainty-principle-limited. In order for tracking to be truly limited by the uncertainty principle, you would have to resolve the track down to the scale at which it receives random kicks from the magnetic field.)
 
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What are expectation values in quantum physics?

Expectation values are a mathematical concept in quantum physics that represent the average value of a physical quantity (such as position or energy) in a given quantum state. They are calculated using the wave function of the system and provide important information about the behavior of quantum systems.

How are expectation values different from definite values in quantum physics?

Definite values, also known as eigenvalues, refer to the specific values that a physical quantity can take on in a given quantum state. Expectation values, on the other hand, represent the average value of that quantity over multiple measurements in that state. While definite values are fixed and unchanging, expectation values can vary depending on the measurement.

Can expectation values be measured in experiments?

Yes, expectation values can be measured in quantum experiments. However, due to the probabilistic nature of quantum systems, the measured expectation value may not always match the theoretical value calculated using the wave function. This is due to the inherent uncertainty in quantum measurements.

How do expectation values relate to the uncertainty principle?

The uncertainty principle, a fundamental principle in quantum physics, states that it is impossible to simultaneously know the exact values of certain pairs of physical quantities (such as position and momentum). Expectation values play a key role in understanding and defining this uncertainty, as they represent the most likely value of a quantity in a given state.

Can expectation values change over time?

Yes, expectation values can change over time in a quantum system. This is because the wave function of a system can evolve over time, leading to different expectation values for the same physical quantity. This is a key feature of quantum mechanics and has important implications for the behavior of quantum systems.

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