Uncertainty Principle in 3 dimensionsI'm trying to understand how to write Heisenberg uncertainty principle in 3 dimensions. What I mean by that is to prove something of the form f ( Δ p x , Δ p y , Δ p z , Δ x , Δ y , Δ z ) ≥ AThis is what I got: The unknown volume that a single particle can be in is Δ V = Δ x Δ y Δ z. The uncertainty in the size of the momentum is Δ p = Δ p x 2 + Δ p y 2 + Δ p z 2 Now this is where I get stuck. In my textbook, for the 1d case, they used De-Broglie equation for connecting the uncertainty of the particle wavelength and its momentum along the x-axis. But does De-Broglie equation is correct per axis or for the size of the vectors?Thanks for you help

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Uncertainty Principle in 3 dimensionsI&#039;m trying to understand how to write Heisenberg uncertainty principle in 3 dimensions. What I mean by that is to prove something of the form   f  (  Δ      p    x    ,  Δ      p    y    ,  Δ      p    z    ,  Δ  x  ,  Δ  y  ,  Δ  z  )  ≥  AThis is what I got: The unknown volume that a single particle can be in is   Δ  V  =  Δ  x  Δ  y  Δ  z. The uncertainty in the size of the momentum is   Δ  p  =      Δ          p      x      2        +    Δ          p      y      2        +    Δ          p      z      2      Now this is where I get stuck. In my textbook, for the 1d case, they used De-Broglie equation for connecting the uncertainty of the particle wavelength and its momentum along the   x-axis. But does De-Broglie equation is correct per axis or for the size of the vectors?Thanks for you help

lavintisqpsnb · Accepted Answer

If you choose any direction in space, the 1D uncertainty principle applies in that direction. So, if you know the component of a particle&#039;s momentum in that direction well, you cannot know its projected position in that direction well. The directions of good and poor localization need not be aligned with your coordinate axes.

Yaritza Oneill · Accepted Answer

There is no need for a 3D uncertainty principle.The operators commute between dimensions:  [                    x        ^              i    ,                    p        ^              j    ]  =  i  ℏ      δ          i      j      Momentum in one dimension and position in another can be measured with arbitrary precision simultaneously (at least there is no restriction from QM).The closest thing I can think of for what you&#039;re asking is:      ∑          i      =      1              3        Δ      x    i    Δ      p    i    ≥      3    2    ℏwhich is not as useful as   Δ      x    i    Δ      p    i    ≥      1    2    ℏ

Uncertainty Principle in 3 dimensions I'm trying to understand how to write Heisenberg uncertainty

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