How are momentum and position related?
In the momentum representation, momentum is the operator of multiplication by p, whereas position is a multiple of differentiation with respect to p. These observables (operators) are not Fourier transforms of each other.
Is momentum dependent on position?
Momentum does not depend on position in classical mechanics either, so it is unclear why it should in QM. Similarly, the rotation operator is defined as ˆR(→α)|→x⟩=|R(→α)→x⟩, where R(→α) denotes a rotation matrix around ˆα with angle |→α|.
What is the position operator in momentum space?
The position operator ˆx, in position space, merely multiplies the wave function by the value x of the position.
How is momentum Fourier transform of position?
Mathematically, the duality between position and momentum is an example of Pontryagin duality. In particular, if a function is given in position space, f(r), then its Fourier transform obtains the function in momentum space, φ(p).
Is momentum independent of position?
They are linearly independent. But the particle’s actual position and actual momentum are not, in general. Unfortunately in physics, we use to mean both the particle trajectory and the coordinate in phase space/real space.
Why do position and momentum not commute?
Momentum Representation The position and momentum operators do not commute in momentum space. It is easy to show that this result is equal to i. The product of the position‐momentum uncertainty is the same in momentum space as it is in coordinate space.
Why is momentum and position independent?
Phase space is defined as the 6 dimensional vector space which contains the position and velocity unit vectors as an orthonormal basis. So in phase space position and velocity are linearly independent (the momentum-position and velocity-position phase spaces are equivalent by scaling).
What are the position operators?
In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. When the position operator is considered with a wide enough domain (e.g. the space of tempered distributions), its eigenvalues are the possible position vectors of the particle.
Can you know the position and momentum of an electron?
The Heisenberg uncertainty principle states that the exact position and momentum of an electron cannot be simultaneously determined. This is because electrons simply don’t have a definite position and direction of motion.
Why is position and velocity independent?
Because at any given instant you can independently control your position and your velocity. Velocity does not depend on position, and position does not depend on velocity. They are independent of each other.
Does momentum commute with position?
Can position and momentum operator have a common Eigenstate?
You can’t. Or rather, you can measure the position, but the result you get will vary from one measurement to the next, because the wavefunction exp(x2/2i−cx) is not an eigenstate of position.