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chrisphd
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Does a position operator exist?
Yes, if the wavefunctions are eigenfunctions of the position operator ;)And if the wavefunction was not an eigenfunction of the position operator, one would expect that the wavefunction will transform into an eigenfunction of the position operator, when operated on.
chrisphd said:And if the wavefunction was not an eigenfunction of the position operator, one would expect that the wavefunction will transform into an eigenfunction of the position operator, when operated on.
No... why would it?Suppose you have a wavefunction that is a linear combination of momentum eigenfunctions. ie, F = 0.8a + 0.6b where F is the wavefunction and a and b are momentum eigenfunctions. When the momentum operator, O, acts on F, the wavefunction F will collapse into either state a, state b with probabilites of 0.64 and 0.36 respectively. So I assumed the same process would occur with the position operator.
No, the momentum operator does not collapse the wavefunction.chrisphd said:Suppose you have a wavefunction that is a linear combination of momentum eigenfunctions. ie, F = 0.8a + 0.6b where F is the wavefunction and a and b are momentum eigenfunctions. When the momentum operator, O, acts on F, the wavefunction F will collapse into either state a, state b with probabilites of 0.64 and 0.36 respectively.
chrisphd said:Mathematically that is what you get when you apply the momentum operator to the wavefunction F. However, I was actually thinking about physically measuring the momentum of the wavefunction, in which the wavefunction then collapses into a or b. So my question is then, when the position of a wavefunction is measured, will the wavefunction likewise collapse into a position eigenfunction just as it would if it was a momentum measurement.
chrisphd said:No... why would it?Suppose you have a wavefunction that is a linear combination of momentum eigenfunctions. ie, F = 0.8a + 0.6b where F is the wavefunction and a and b are momentum eigenfunctions. When the momentum operator, O, acts on F, the wavefunction F will collapse into either state a, state b with probabilites of 0.64 and 0.36 respectively. So I assumed the same process would occur with the position operator.
I think you are mixing things up here. The action of a measurement is not given by acting with an operator on the state. The simplest interpretation of a measurement is that the |coefficients|^2 represent the probability of a certain outcome, and that after that measurement the wavefunction has collapsed. But that does not involve the action of the operator itself. The action of any (physical) operator on a wavefunction is never a collapse. This also goes for the momentum operator.
A position operator is a mathematical operator used in quantum mechanics to describe the position of a particle in space. It represents the location of a particle in three-dimensional space and is denoted by the symbol x.
No, a position operator does not exist in classical mechanics. In classical mechanics, the position of a particle is described by its coordinates in space, whereas in quantum mechanics, the position is described by the position operator which is a mathematical abstraction.
The position operator is related to the uncertainty principle, which states that it is impossible to simultaneously determine the position and momentum of a particle with absolute certainty. The position operator plays a crucial role in this principle, as it represents the measurement of position, which can never be known with complete accuracy.
No, the position operator itself cannot be observed or measured directly. It is a mathematical concept used in quantum mechanics to describe the position of a particle. However, the position of a particle can be measured experimentally using physical instruments and techniques.
The position operator is used in quantum mechanics to represent the position of a particle in three-dimensional space. It is an essential part of the mathematical framework used to describe the behavior of particles at the quantum level. The position operator is also used to calculate the probability of finding a particle at a certain location in space.