Quantum state with well-defined position and momentum?

In summary: The Heisenberg uncertainty principle is a fundamental principle that arises from the non-commutativity of position and momentum operators. It states that the product of the uncertainties in measuring position and momentum must always be greater than or equal to a certain value. This means that there is no way to know both the exact position and momentum of a particle at the same time. This principle is not related to measurement, but rather a mathematical consequence of the properties of quantum systems. In summary, the conversation discussed the concept of a simultaneous eigenstate for position and momentum and whether it exists or not. It was concluded that such an eigenstate does not exist due to the non-commutativity of the position and momentum operators, which is a fundamental principle in quantum
  • #1
romsofia
597
310
Well, I know if both the position and momentum are in a simultaneous eigenstate then, theoretically, we would be able to measure momentum and position without changing the wavefunction. But, is there such an eigenstate out there?

Any help is appreciated!

(sorry if the wording is awkward)
 
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  • #2
romsofia said:
is there such an eigenstate out there?

No, because of the Heisenberg uncertainty principle.
 
  • #3
jtbell said:
No, because of the Heisenberg uncertainty principle.

I've always thought they're would be maybe ONE eigenstate which could contain them both.

I just found a site proving it though: http://inst.eecs.berkeley.edu/~cs191/fa05/lectures/lecture13_fa05.pdf
(I never saw a proof, guess I have a flaw in my understanding of the principle!).

Thanks for answering though!
 
  • #4
The eigenstates for position and momentum are noncommutative, hence the uncertainty principle.
 
  • #5
What Chronos meant is that the position and momentum operators are non-commutative, therefore you cannot construct a simultaneous eigenstate.

One should mention that this is true only for position and momentum in the same direction.
 
  • #6
There is a general misconception that this has something to do with measurement; that's not true. The HUP followes as a strict mathematical theorem from Hilbert space geometry. This demonstrates that for non-commuting operators the product of their uncertainties is always non-zero. This holds for arbitrary states, i.e. there are states minimizing the HUP, but there are no states for which the result is exactly zero!
 
  • #7
romsofia said:
Well, I know if both the position and momentum are in a simultaneous eigenstate then, theoretically, we would be able to measure momentum and position without changing the wavefunction. But, is there such an eigenstate out there?

Any help is appreciated!

(sorry if the wording is awkward)

There's no common eigenstate. Not even one. If it were, it would violate the canonical commutation relations which are the cornerstone of quantum mechanics.
 

Related to Quantum state with well-defined position and momentum?

1. What is a quantum state with well-defined position and momentum?

A quantum state with well-defined position and momentum is a state in which both the position and momentum of a particle are known with certainty. This means that the probability of finding the particle at a specific position and with a specific momentum is equal to 100%.

2. How is a quantum state with well-defined position and momentum different from other quantum states?

Unlike other quantum states, which have a level of uncertainty in either the position or momentum of a particle, a quantum state with well-defined position and momentum has no uncertainty and is considered a "pure" state.

3. Can a quantum state with well-defined position and momentum exist in real life?

Yes, a quantum state with well-defined position and momentum can exist in real life, but only under very specific circumstances. One example is the ground state of a hydrogen atom, where the electron's position and momentum are both known with certainty.

4. How is a quantum state with well-defined position and momentum described mathematically?

A quantum state with well-defined position and momentum is described by a wavefunction, which is a mathematical representation of the particle's position and momentum. The wavefunction is a complex-valued function that contains all the information about the particle's quantum state.

5. What is the Heisenberg uncertainty principle and how does it relate to a quantum state with well-defined position and momentum?

The Heisenberg uncertainty principle states that it is impossible to know both the position and momentum of a particle with absolute certainty. Therefore, a quantum state with well-defined position and momentum violates this principle, as it implies that both quantities are known with certainty. This is only possible in specific situations and cannot be achieved simultaneously for all particles.

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