We know, Magnetic vector potential plays the crucial part in the Hamiltonian of an Electromagnetic system where the Hamiltonian formulation is the basis definition of transformation of equations from classical to Quantum.
And Experimental results like Aharonov-Bohm effect make it look like the magnetic vector potential is inevitable in Quantum mechanics. Together, they state the importance of vector potential
But, if we consider the possibility of monopoles, then we may have to reject the concept of vector potential and need to redefine our Hamiltonian with new potentials, which in turn may result into contradictions with unexpected results.
To prevent this, we reject at the beginning itself the possibility of an isolated magnetic monopole in our Universe, so that everyone can live in their happy little world with conventional equations.
But, Dirac first took a completely different approach with his new mathematical treatment along with vector potential and proposed a new magnetic monopole that can even co-exist with the vector potential. i.e.without any change in our old potential formalism.
Moreover, the vector potential itself allows for a such particle to exist in nature without any violations.
Let us look at that approach from his 1931 paper..
First we introduce the wave function in the usual form as, $$ \vert\psi\rangle = Ae^{i\gamma} $$ where $\gamma$ is the function of x,y,z,t and also we take A - amplitude as the function of position and time since we take the general case.
Now, the indeterminacy in this wave function can be regarded as the possible addition of any constant to the phase. That is equivalent to $$\psi = A e^{i\gamma + \chi} = Ae^{i\gamma} e^{\chi}$$ where $\chi$ is some constant. It doesn't change anything physical about the wave function, because we know that from superposition, the wave function will not change from the possible multiplication of any arbitrary real or complex number.
So, you can never determine any wave function up to an arbitary constant which can be chosen arbitrarily to normalize the wave function.
From this fact, the wave function cannot have a definite phase at all the points but can have only definite difference. All it does matter is the difference in $\gamma$ value. But we are not sure whether this difference is unique for any two arbitrary points, as there are many paths from going from one point to another. We are not even sure whether the closed integral of this phase change will vanish or not.
Away from this, the definitions of our phase change in any sense should not give rise to ambiguity in the applications of the theory.
First it is seen that the concept of phase doesn't change anything in the density function as, $$ \langle\psi\vert\psi\rangle = A^2 e^{i\gamma} e^{-i\gamma} = A^2 $$ which is a pure real number that doesn't depend on the value of phase (as the phase vanishes by its complex conjugate).
But, it is no longer the case if we take two different wavefunctions.
$$ \langle\psi_m\vert\psi_n\rangle = \langle\psi_m\vert[c_1\psi_1 + c_2\psi_2 + ...+ c_m\psi_m + ...]\rangle = c_m $$
where I expanded $\psi_n$ in terms of the eigen functions $\psi_m$.
We know from the Quantum Mechanics postulates that, $\vert{c_m}\vert^2$ gives the probability of $\psi_m$ state in $\psi_n $ state which termed by dirac as, probability of agreement of the two states.
The limits of the integral in the bra-ket notation needn't to be from $(-\infty,\infty)$.
I think, this may be the only disadvantage in Dirac notation. The limits are not represented explicitly.
Since the integral does depend on the end points, even though the wave functions don't have any definite phase, they should have definite phase difference between two points (because the integral is a number).
The same physical argument gives us that, the change in phase round the closed path should be zero.
Let us look at it like this by saying,
the phase of the wave function $\psi_m$ is $\gamma_m$ and for the second $\psi_n$ is $\gamma_n$ So, the phase of $ \langle\psi_m\vert\psi_n\rangle $ is $e^{i(\gamma_n - \gamma_m)}$
When we go along the path from one point to another, there is a corresponding change in phase for each wave function respectively $\chi_m\,,\,\chi_n$.
And so, the new phase difference at this point is $$ e^{i(\gamma'_n -\gamma'_m)} = e^{i[(\gamma_n +\chi_n)-(\gamma_m+\chi_m)]}$$ For different set of two points, the phase difference can have definite values. But, we know that from the physical fact that, if we come again at the same initial point, we should have the same probability and so the same integral value on closed path integral, i.e. $$ e^{i(\gamma'_n-\gamma'_m)} = e^{i(\gamma_n -\gamma_m)} $$ or $$ \gamma_n + \chi_n -\gamma_m-\chi_m = \gamma_n - \gamma_m \\ \rightarrow \chi_n = \chi_m $$
Which says that the change in phase of $\psi_m$ and $\psi_n$ should be the same and opposite around a closed path.
Since it is a general result, it can be stated as in Dirac's paper,
The change in phase of a wave function round any closed curve must be the same for all the wave functions.
The change in phase doesn't talk anything about the nature of wave function or concerned with any specific system. So, the change in phase should be something a property of the dynamical system or the force field in which it moves.
For the mathematical treatment, it was expressed the wave function as, $$ \psi = \psi_1 e^{i\beta}$$ $\psi_1$ being the usual wave function with definite phase and the uncertainty in phase is put in the factor $ e^{i\beta}$ where $beta$ is the same as $\chi$ we used in the previous. This $\beta$ having definite values at each point is not a function of x,y,z,t. (because different paths at the same point will possibly give different values of phase change). But it has definite derivatives at each point (x,y,z,t).
We represent its derivatives as, $$ k_x =\frac{\partial\beta}{\partial{x}} \\ k_y =\frac{\partial\beta}{\partial{y}}\\k_z =\frac{\partial\beta}{\partial{z}}\\k_0 =\frac{\partial\beta}{\partial{t}}$$ In general these derivatives needn't be integrable following the condition $$ \frac{\partial\beta}{\partial{y}\partial{x}} = \frac{\partial\beta}{\partial{x}\partial{y}}$$
Now, using Stokes' theorem, we try to calculated the change in phase around a closed path as, $$ \oint \vec{K}\cdot\vec{dl} = \int (\nabla\times\vec{K})\cdot\vec{dS} $$
where the length and area element is considered in four dimensions, since K has four components.
Here is one essential point we shouldn't assume, that is all the wave functions should have the same phase factor $e^{i\beta}$ because of the fact that all wave functions have same phase difference along a closed path.
The reason is because only the change in phase depend on the curl of K vector. We can still change the components by the gradient of any scalar function, so that same phase difference can be obtained to be the same.
We can start from here in the next post.
And Experimental results like Aharonov-Bohm effect make it look like the magnetic vector potential is inevitable in Quantum mechanics. Together, they state the importance of vector potential
But, if we consider the possibility of monopoles, then we may have to reject the concept of vector potential and need to redefine our Hamiltonian with new potentials, which in turn may result into contradictions with unexpected results.
To prevent this, we reject at the beginning itself the possibility of an isolated magnetic monopole in our Universe, so that everyone can live in their happy little world with conventional equations.
But, Dirac first took a completely different approach with his new mathematical treatment along with vector potential and proposed a new magnetic monopole that can even co-exist with the vector potential. i.e.without any change in our old potential formalism.
Moreover, the vector potential itself allows for a such particle to exist in nature without any violations.
Let us look at that approach from his 1931 paper..
First we introduce the wave function in the usual form as, $$ \vert\psi\rangle = Ae^{i\gamma} $$ where $\gamma$ is the function of x,y,z,t and also we take A - amplitude as the function of position and time since we take the general case.
Now, the indeterminacy in this wave function can be regarded as the possible addition of any constant to the phase. That is equivalent to $$\psi = A e^{i\gamma + \chi} = Ae^{i\gamma} e^{\chi}$$ where $\chi$ is some constant. It doesn't change anything physical about the wave function, because we know that from superposition, the wave function will not change from the possible multiplication of any arbitrary real or complex number.
So, you can never determine any wave function up to an arbitary constant which can be chosen arbitrarily to normalize the wave function.
From this fact, the wave function cannot have a definite phase at all the points but can have only definite difference. All it does matter is the difference in $\gamma$ value. But we are not sure whether this difference is unique for any two arbitrary points, as there are many paths from going from one point to another. We are not even sure whether the closed integral of this phase change will vanish or not.
Away from this, the definitions of our phase change in any sense should not give rise to ambiguity in the applications of the theory.
First it is seen that the concept of phase doesn't change anything in the density function as, $$ \langle\psi\vert\psi\rangle = A^2 e^{i\gamma} e^{-i\gamma} = A^2 $$ which is a pure real number that doesn't depend on the value of phase (as the phase vanishes by its complex conjugate).
But, it is no longer the case if we take two different wavefunctions.
$$ \langle\psi_m\vert\psi_n\rangle = \langle\psi_m\vert[c_1\psi_1 + c_2\psi_2 + ...+ c_m\psi_m + ...]\rangle = c_m $$
where I expanded $\psi_n$ in terms of the eigen functions $\psi_m$.
We know from the Quantum Mechanics postulates that, $\vert{c_m}\vert^2$ gives the probability of $\psi_m$ state in $\psi_n $ state which termed by dirac as, probability of agreement of the two states.
The limits of the integral in the bra-ket notation needn't to be from $(-\infty,\infty)$.
I think, this may be the only disadvantage in Dirac notation. The limits are not represented explicitly.
Since the integral does depend on the end points, even though the wave functions don't have any definite phase, they should have definite phase difference between two points (because the integral is a number).
The same physical argument gives us that, the change in phase round the closed path should be zero.
Let us look at it like this by saying,
the phase of the wave function $\psi_m$ is $\gamma_m$ and for the second $\psi_n$ is $\gamma_n$ So, the phase of $ \langle\psi_m\vert\psi_n\rangle $ is $e^{i(\gamma_n - \gamma_m)}$
When we go along the path from one point to another, there is a corresponding change in phase for each wave function respectively $\chi_m\,,\,\chi_n$.
And so, the new phase difference at this point is $$ e^{i(\gamma'_n -\gamma'_m)} = e^{i[(\gamma_n +\chi_n)-(\gamma_m+\chi_m)]}$$ For different set of two points, the phase difference can have definite values. But, we know that from the physical fact that, if we come again at the same initial point, we should have the same probability and so the same integral value on closed path integral, i.e. $$ e^{i(\gamma'_n-\gamma'_m)} = e^{i(\gamma_n -\gamma_m)} $$ or $$ \gamma_n + \chi_n -\gamma_m-\chi_m = \gamma_n - \gamma_m \\ \rightarrow \chi_n = \chi_m $$
Which says that the change in phase of $\psi_m$ and $\psi_n$ should be the same and opposite around a closed path.
Since it is a general result, it can be stated as in Dirac's paper,
The change in phase of a wave function round any closed curve must be the same for all the wave functions.
The change in phase doesn't talk anything about the nature of wave function or concerned with any specific system. So, the change in phase should be something a property of the dynamical system or the force field in which it moves.
For the mathematical treatment, it was expressed the wave function as, $$ \psi = \psi_1 e^{i\beta}$$ $\psi_1$ being the usual wave function with definite phase and the uncertainty in phase is put in the factor $ e^{i\beta}$ where $beta$ is the same as $\chi$ we used in the previous. This $\beta$ having definite values at each point is not a function of x,y,z,t. (because different paths at the same point will possibly give different values of phase change). But it has definite derivatives at each point (x,y,z,t).
We represent its derivatives as, $$ k_x =\frac{\partial\beta}{\partial{x}} \\ k_y =\frac{\partial\beta}{\partial{y}}\\k_z =\frac{\partial\beta}{\partial{z}}\\k_0 =\frac{\partial\beta}{\partial{t}}$$ In general these derivatives needn't be integrable following the condition $$ \frac{\partial\beta}{\partial{y}\partial{x}} = \frac{\partial\beta}{\partial{x}\partial{y}}$$
Now, using Stokes' theorem, we try to calculated the change in phase around a closed path as, $$ \oint \vec{K}\cdot\vec{dl} = \int (\nabla\times\vec{K})\cdot\vec{dS} $$
where the length and area element is considered in four dimensions, since K has four components.
Here is one essential point we shouldn't assume, that is all the wave functions should have the same phase factor $e^{i\beta}$ because of the fact that all wave functions have same phase difference along a closed path.
The reason is because only the change in phase depend on the curl of K vector. We can still change the components by the gradient of any scalar function, so that same phase difference can be obtained to be the same.
We can start from here in the next post.
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