Is the reason that knowing the position of an electron doesn't allow you to know about it's momentum because the position of an electron is based on probability, so since it is impossible to know it's second position, it is impossible to determine it's velocity and thus, it's momentum?
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Problem 1 - If the photon has a short wavelength, and therefore a large momentum, the position can be measured accurately. But the photon scatters in a random direction, transferring a large and uncertain amount of momentum to the electron. If the photon has a long wavelength and low momentum, the collision does not disturb the electron's momentum very much, but the scattering will reveal its position only vaguely.
Problem 2 - If a large aperture is used for the microscope, the electron's location can be well resolved (see Rayleigh criterion); but by the principle of conservation of momentum, the transverse momentum of the incoming photon and hence the new momentum of the electron resolves poorly. If a small aperture is used, the accuracy of both resolutions is the other way around.
The combination of these trade-offs imply that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower limit, which is (up to a small numerical factor) equal to Planck's constant.
Problem 2 - If a large aperture is used for the microscope, the electron's location can be well resolved (see Rayleigh criterion); but by the principle of conservation of momentum, the transverse momentum of the incoming photon and hence the new momentum of the electron resolves poorly. If a small aperture is used, the accuracy of both resolutions is the other way around.
The combination of these trade-offs imply that no matter what photon wavelength and aperture size are used, the product of the uncertainty in measured position and measured momentum is greater than or equal to a lower limit, which is (up to a small numerical factor) equal to Planck's constant.
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You are of the opinion that velocity is determined only if one knows the second position.
If one knows precisely a position, its velocity at that point is also precisely known.
But the fact is that one cannot precisely know the position of a particle because of the wave nature of the particle. What we know is the probability of its position. Being there an uncertainty about the position, there is uncertainty in finding its velocity.
In short, the first position and first velocity are uncertain.
Similarly there is uncertainty in its second position and second velocity.
If one knows precisely a position, its velocity at that point is also precisely known.
But the fact is that one cannot precisely know the position of a particle because of the wave nature of the particle. What we know is the probability of its position. Being there an uncertainty about the position, there is uncertainty in finding its velocity.
In short, the first position and first velocity are uncertain.
Similarly there is uncertainty in its second position and second velocity.
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To know where something is you have to "see" it.
This involves reflecting a photon from it.
The photon contains momentum and changes the momentum of the electron.
The photon has a wavelength and as such isn't confined to a point so you don't know EXACTLY where IT is, but the higher the momentum the smaller the wavelength of the photon and the more precisely you know its position.
However the higher momentum means more uncertainty in the final momentum of the electron.
So you just can't win. That is the uncertainty principle.
This involves reflecting a photon from it.
The photon contains momentum and changes the momentum of the electron.
The photon has a wavelength and as such isn't confined to a point so you don't know EXACTLY where IT is, but the higher the momentum the smaller the wavelength of the photon and the more precisely you know its position.
However the higher momentum means more uncertainty in the final momentum of the electron.
So you just can't win. That is the uncertainty principle.
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Ask Heisenberg - it's his Law. (Well, maybe I'm a little uncertain - could be just my principles)
(Sorry. I never know what direction I'll go with my answers when I'm at my keyboard.) :)
(Sorry. I never know what direction I'll go with my answers when I'm at my keyboard.) :)