{wiki}

This operator case is surprisingly not necessarily mathematically trivial to describe formally because you often end up getting into the <Dirac delta functions>/continuous spectrum: as mentioned at: <mathematical formulation of quantum mechanics>


Position operator

{wiki}

One dimension in <position representation>:
$$
\hat{p} = - i \hbar \pdv{}{x}
$$

In three dimensions In position representation, we define it by using the <gradient>, and so we see that 
$$
\hat{p} = - i \hbar \pdv{}{x}
$$

\Video[https://www.youtube.com/watch?v=Egu4i8umpoM]
{title=Position and Momentum from Wavefunctions by <Faculty of Khan> (2018)}
{description=Proves in detail why the <momentum operator> is $\pdv{}{x}$. The starting point is determining the time <derivative> of the <expectation value> of the <position operator>.}


Momentum operator

Squeezed coherent state

Wien approximation

{c}
{title2=1900}
{wiki}

Derived <from first principles>[from classical first principles], matches <Planck's law> for low frequencies, but diverges at higher frequencies.


Rayleigh-Jeans law

Integrated circuit layout

{tag=Organization developing quantum hardware}


Organization developing photonic quantum computer

Extreme ultraviolet lithography

Photomask

{c}
{wiki=Myers–Briggs_Type_Indicator}


Myers-Briggs Type_Indicator

Big Five personality traits

Slater determinant

Fermion

{wiki=Spin–statistics_theorem}

<video The Biggest Ideas in the Universe | 17. Matter by Sean Carroll (2020)> at https://youtu.be/dQWn9NzvX4s?t=3707 says that no one has ever been able to come up with an intuitive reason for the proof.


Spin-statistics theorem

Electron degeneracy pressure

As mentioned at: https://physics.stackexchange.com/questions/212726/a-quantum-particle-moving-from-a-to-b-will-take-every-possible-path-from-a-to-b/212790#212790[], classical <Gravity waves> for example also "take all possible paths". This is just what waves look like they are doing.


Quantum particles take all possible paths

Propagator

Thought experiment that illustrates the <path integral formulation> of <quantum field theory>.

Mentioned for example in <quantum field theory in a nutshell by Anthony Zee (2010)> page 8.


Ciro Santilli @cirosantilli  40