Proton Mixed A and S wavefunctions charge

In summary, the conversation discusses proton wavefunctions of mixed symmetry and their expectation values for the charge operator. The charge of the proton is defined as ##Q = q_1 + q_2 + q_3##, not ##q_3##, and the expectation values for the charge operators on the wavefunctions |MA> and |MS> are ##<MA|q_3|MA>=\frac{2}{3}## and ##<MS|q_3|MS>=0##, respectively. This shows that the charge of |MA> is ##q_{MA}=\frac{2}{3}## and the charge of |MS> is ##q_{MS}=0##, which is not the
  • #1

ChrisVer

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I have the proton wavefunctions of mixed symmetry:

[itex] |MA> = \frac{1}{\sqrt{2}} ( u_1 d_2 u_3 - d_1 u_2 u_3 ) [/itex]

and

[itex] |MS> = \frac{1}{\sqrt{6}} (2u_1 u_2 d_3 - u_1 d_2 u_3 - d_1u_2u_3 ) [/itex]

If the charge defined as: [itex]qu =\frac{2}{3} u [/itex] and [itex]qd= -\frac{1}{3} d [/itex] , I need to show what are the expectation values of [itex]q_3[/itex] acting on the third particle:

[itex] <MA| q_3 | MA> ~~,~~ <MS|q_3|MS> [/itex]

However I get [itex]q_{MA}= \frac{2}{3} [/itex] and [itex]q_{MS}=0[/itex]. Is that rational? None of these are the proton's charge...

For MA it's easy to see:

[itex] q_3 |MA> = \frac{1}{\sqrt{2}} \Big[ q_3 (u_1d_2u_3) - q_3 (d_1u_2u_3 ) \Big] = \frac{1}{\sqrt{2}} \Big[ \frac{2}{3} (u_1d_2u_3) - \frac{2}{3} (d_1u_2u_3 ) \Big] = \frac{2}{3} |MA> [/itex]

So that: [itex] <MA|q_3 |MA> = \frac{2}{3} [/itex]

In a similar but a bit more complicated way (since |MS> doesn't appear to be an eigenstate) I calculated that ##q_{MS}= \frac{1}{6} [ -\frac{4}{3} + \frac{2}{3} + \frac{2}{3}]=0## ...
However I don't understand why I don't get =+1
 
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  • #2
The charge of the proton is not ##q_3##, it is ##Q = q_1 + q_2 + q_3## ...
 

What is a proton?

A proton is a subatomic particle that is found in the nucleus of an atom. It has a positive charge and is approximately 1,836 times more massive than an electron.

What is a mixed A and S wavefunction?

A mixed A and S wavefunction is a mathematical representation of the energy and spatial distribution of a proton. It takes into account both the angular momentum (S wave) and the radial motion (A wave) of the proton.

What is the charge of a proton?

The charge of a proton is positive, with a value of +1.602 x 10^-19 Coulombs. This charge is equal and opposite to the charge of an electron, which is negative.

How do A and S wavefunctions affect the charge of a proton?

The A and S wavefunctions do not directly affect the charge of a proton. However, they do play a role in determining the overall energy and behavior of the proton, which can indirectly impact its charge in certain situations.

Why is understanding proton mixed A and S wavefunctions important?

Understanding proton mixed A and S wavefunctions is important because it allows us to better understand the structure and behavior of protons, which are essential building blocks of matter. This knowledge can also help us make predictions and calculations in fields such as quantum mechanics and nuclear physics.

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