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Could you tell me if I have understood following about operators in QM correctly?

Wavefunction takes all generalized coordinates of the system as arguments.

for example if we have a system of proton and electron (in 3-dimensional space) then the wavefunction of this system has 7 arguments ##\psi_{example}(t, x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})##. in this example generalized coordinates of the system are normal coordinates of pointbodies of the system.

In example-system: to find probability of proton being in on coordinate ##(43,21,65)## and electron being in coordinates (43,2,12) at time 5 is ##P(5, 43,21,65,43,2,12)=|\psi_{example}(5, 43,21,65,43,2,12)|^2##

operator applied to a wavefunction equals to wavefunction times the value that the operator is measuring. ##\hat{O}(\psi)(arguments\ of\ psi)=\psi(arguments\ of\ psi)\ \lambda(arguments\ of\ psi)## .

for example: if in the example-system I apply momentum operator to get projection of momentum of electron to the first coordinate direction, then ##\hat{O}_{get\ x-momentum\ of\ electron}(\psi_{example})=-i\ h/(2pi)\ \frac{\partial \psi_{example}}{\partial x_{1.1}}=p_{1.1}\ \psi=m_{electron}\ v_{1.1}\ \psi##

this equation can be simplified to ##\hat{O}(\psi)(arguments\ of\ psi)/\psi(arguments\ of\ psi)=\lambda(arguments\ of\ psi)## where lambda is the quantity, that the operator is measuring.

The equation about the example-system can be simplified to ##p_{1.1}(t,x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})=-\frac{i\ h}{2pi\ \psi_{example}(t,x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})}\ \frac{\partial \psi_{example}}{\partial x_{1.1}}(t,x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})## .

If ##\psi## is eigenfunction of operator ##\hat{O}## then ##\lambda## is the eigenvalue and does not depend of arguments of ##\psi## and the quantity that the ##\hat{O}## measures is same in every time and every place in the system.

For example if projection of momentum of the electron to the 1. coordinate is known and does not depend of place and time. Then ##p_{1.1}## does not depend of ##x_{0.1}## , ##x_{0.2}## , ##x_{0.3}## , ##x_{1.1}## , ##x_{1.2}## nor ##x_{1.3}##. It can be so only if wavefunction ##\psi_{example}##can be written in the form ##\psi_{example}(t,x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})=F(t,x_{0.1},x_{0.2},x_{0.3},x_{1.2},x_{1.3})\ e^{C*x_{1.1}}## where F is an arbitary function and C is an arbitary compleksnumber. In this case the electron is equally likely to have any first-coordinate value.

If ##\psi## is not eigenfunction of ##\hat{O}## then the value, that the operator ##\hat{O}## measures, does depend of generalized coordinates of the system.

for example in example-system momentum of the electron

##p_{1.1}(t,x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})=-\frac{i\ h}{2pi\ \psi_{example}(t,x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})}\ \frac{\partial \psi_{example}}{\partial x_{1.1}}(t,x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})## does depend of time and coordinates of proton and electron.

Wavefunction takes all generalized coordinates of the system as arguments.

for example if we have a system of proton and electron (in 3-dimensional space) then the wavefunction of this system has 7 arguments ##\psi_{example}(t, x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})##. in this example generalized coordinates of the system are normal coordinates of pointbodies of the system.

- ##t## is time
- ##x_{0.1}## is projection of position of proton to 1.-coordinate-axis.
- ##x_{0.2}## is projection of position of proton to 2.-coordinate-axis.
- ##x_{0.3}## is projection of position of proton to 3.-coordinate-axis.
- ##x_{1.1}## is projection of position of electron to 1.-coordinate-axis.
- ##x_{1.2}## is projection of position of electron to 2.-coordinate-axis.
- ##x_{1.3}## is projection of position of electron to 3.-coordinate-axis.

In example-system: to find probability of proton being in on coordinate ##(43,21,65)## and electron being in coordinates (43,2,12) at time 5 is ##P(5, 43,21,65,43,2,12)=|\psi_{example}(5, 43,21,65,43,2,12)|^2##

operator applied to a wavefunction equals to wavefunction times the value that the operator is measuring. ##\hat{O}(\psi)(arguments\ of\ psi)=\psi(arguments\ of\ psi)\ \lambda(arguments\ of\ psi)## .

for example: if in the example-system I apply momentum operator to get projection of momentum of electron to the first coordinate direction, then ##\hat{O}_{get\ x-momentum\ of\ electron}(\psi_{example})=-i\ h/(2pi)\ \frac{\partial \psi_{example}}{\partial x_{1.1}}=p_{1.1}\ \psi=m_{electron}\ v_{1.1}\ \psi##

this equation can be simplified to ##\hat{O}(\psi)(arguments\ of\ psi)/\psi(arguments\ of\ psi)=\lambda(arguments\ of\ psi)## where lambda is the quantity, that the operator is measuring.

The equation about the example-system can be simplified to ##p_{1.1}(t,x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})=-\frac{i\ h}{2pi\ \psi_{example}(t,x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})}\ \frac{\partial \psi_{example}}{\partial x_{1.1}}(t,x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})## .

If ##\psi## is eigenfunction of operator ##\hat{O}## then ##\lambda## is the eigenvalue and does not depend of arguments of ##\psi## and the quantity that the ##\hat{O}## measures is same in every time and every place in the system.

For example if projection of momentum of the electron to the 1. coordinate is known and does not depend of place and time. Then ##p_{1.1}## does not depend of ##x_{0.1}## , ##x_{0.2}## , ##x_{0.3}## , ##x_{1.1}## , ##x_{1.2}## nor ##x_{1.3}##. It can be so only if wavefunction ##\psi_{example}##can be written in the form ##\psi_{example}(t,x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})=F(t,x_{0.1},x_{0.2},x_{0.3},x_{1.2},x_{1.3})\ e^{C*x_{1.1}}## where F is an arbitary function and C is an arbitary compleksnumber. In this case the electron is equally likely to have any first-coordinate value.

If ##\psi## is not eigenfunction of ##\hat{O}## then the value, that the operator ##\hat{O}## measures, does depend of generalized coordinates of the system.

for example in example-system momentum of the electron

##p_{1.1}(t,x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})=-\frac{i\ h}{2pi\ \psi_{example}(t,x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})}\ \frac{\partial \psi_{example}}{\partial x_{1.1}}(t,x_{0.1},x_{0.2},x_{0.3},x_{1.1},x_{1.2},x_{1.3})## does depend of time and coordinates of proton and electron.

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