Before stating the postulate, there is a special property we should know about the eigenfunctions corresponding to the distinct eigen values.
It is an easy one to derive. Let us consider two eigenstates with distinct eigen values i.e. eigen states $ \vert\phi_1\rangle $ and $\vert\phi_2\rangle $ with the corresponding eigen values $\lambda_1 $ and $\lambda_2$.
From eq.(7) in the part-1, we use the definition of Hermitian Operator, $$ \int \phi_1^* (\hat{Q} \phi_2) \,dx = \int (\hat{Q}\phi_1)^*\phi_2 \,dx $$
Note: The equation applies for any two function because of the definition of Hermiticity (of an Operator).
Using eq.(5) - eigen value equation
$$ \int \phi_1^* (\lambda_2\phi_2) \,dx = \int (\lambda_1^* \phi_1^*) \phi_2 \,dx $$
Since $\lambda_1\, , \, \lambda_2 $ are eigen values, they are real numbers. So, $$ \lambda_1^* = \lambda_1 $$
$$ \lambda_2 \int \phi_1^* \phi_2 \,dx = \lambda_1 \int \phi_1^* \phi_2 \,dx $$
which gives that,
$$ (\lambda_2 - \lambda_1) \int \phi_1^* \phi_2 \,dx = 0 $$
We assumed that eigenvalues are distinct, so $$ \lambda_2 - \lambda_1 \neq 0 $$
The only possibility is that, $$ \int \phi_1^* \phi_2 \,dx = 0 \,\,\,\,\, \ldots... eq.(8)$$
It is possible only when the eigen functions are orthogonal to each other. Thus we get that, the eigenfunctions corresponding to distinct eigenvalues are orthogonal to each other. And it is always possible to make the eigenfunctions as the basis for our Linear vector Space (even when there are same eigen values for different eigenfunctions - it is called degeneracy. Those cases should be dealt separately).
Now, we will see our fourth postulate, that is, "The number of measurement that will result into the particular eigenvalue is proportional to the square of the magnitude of the coefficent of that particular eigenfunction in the expansion of the wave function.
When, it is said "the expansion", remember how the wave function (Ket vector) was written in terms of the basis vectors in eq.(1).
But now, we have learnt that, "Eigen functions are always the one of the possible set of basis vectors" and so, the wave function in general can always be expanded in terms of these new eigen functions as the basis (similar to eq.(1)),
$$ \vert\psi(t)\rangle = A_1(t) \vert\phi_1\rangle + A_2(t) \vert\phi_2\rangle + ...$$
Where $\vert\phi_1\rangle , \vert\phi_2\rangle $ are eigen functions and let us say their eigen values are $\lambda_1, \lambda_2 $.
Now, the postulates interprets that, the probability of a particular eigen value $\lambda_1 or \lambda_2$ is proportional to the square of the magnitude of the coefficient of the eigen function in the expansion of wave function i.e. $A_1 $ and $ A_2 $
where, $ A_1^2 $ gives the probability of the eigen value $\lambda_1 $ with the eigen function $\phi_1$ and similarly for others.
Using inner product, the value of ,
$$ A_1^2 = |\langle\phi_1\vert\psi(t)\rangle|^2 $$
In general,
$$ A_b^2 = |\langle\phi_b\vert\psi(t)\rangle|^2 \,\,\,\, \ldots...eq.(9)$$
This is the more general Statement of Max Born's interpretation.
There are many things you can predict with this postulate and it plays the most significant role hereafter.
When, I say eigen functions, there are really infinite possible sets of eigen functions you can choose in Nature. Depending on the problems, we will mostly used to deal with Energy, Position and Momentum Eigen functions.
In the beginning itself, we mentioned.. dealing with abstract ket vectors is not possible and so it should be taken the projection of these ket vectors in a desired function space.
In particular, when I take the projection on position space, we get Position Space wave function expressed as the linear combination of eigen function in position space as, from eq.(2),
$$ \langle{x}\vert\psi(t)\rangle = \sum_b A_b(t) \langle{x}\vert\phi_b\rangle \,\,\,\, \ldots...eq.(2)$$
$$ \psi(x,t) = \sum_b A_b(t) \phi_b(x) $$
where $\phi_b(x) $ is some general eigen function in position space.
But, if the eigen functions itself are position eigen functions?
Then, the inner product $\langle{x}\vert\psi(t) $ becomes dirac delta function, which means, it gives the value "one" only at that specific "x" and gives the value zero for all other x.
The reason is, when you are in one eigen state, you can only measure the corresponding eigen value and not some other eigen value of some different eigen function.
Therefore, eq.(2) becomes,
$$ \langle{x}\vert\psi(t)\rangle = \sum_b A_b(t) \langle{x}\vert{x'}\rangle = \sum_b A_b(t) \delta(x-x') $$
Thus, we obtain the value of probability amplitude,
$$ \psi(x,t) = A_b(t) $$ and $$ |\psi(x,t)|^2 = A_b^2 \,\,\,\, \ldots...eq.(10)$$
where $A_b^2 $ is the probability of getting the eigen value corresponding to eigen function $\vert{x}\rangle $.
Equating with the left side, we get the general property of a "Position space Wavefunction" and that is "the square of the amplitude of the Position space wave function gives the probability of finding the particle at x and x+dx".
This is popularly known as the Max Born Interpretation in the basic Quantum Mechanics.
Just because we found a meaning for the $ |\psi(x,t)|^2$ doesn't mean, we can only define it in terms of position eigen functions.
Still, we could expand this "Position space wave function in terms of Energy eigen functions or Momentum Eigenfunctions or anything, but in Position space (means Position as the parameter).
Instead of Position space, we can also take the inner product with momentum and get momentum space wave function and can define similar things. In that case,
$$ \langle{p}\vert\psi(t)\rangle = \psi(p,t) $$
and $$ |\psi(p,t)|^2 $$ gives the probability of finding the particle at momentum p and p+dp.
All you need to understand is the Vector space and its basis.
For a vague imagination, you can compare it with various coordinates like explaining a vector in terms of cartesian or spherical or cylindrical or etc. But, it is just for an understanding.
It is an easy one to derive. Let us consider two eigenstates with distinct eigen values i.e. eigen states $ \vert\phi_1\rangle $ and $\vert\phi_2\rangle $ with the corresponding eigen values $\lambda_1 $ and $\lambda_2$.
From eq.(7) in the part-1, we use the definition of Hermitian Operator, $$ \int \phi_1^* (\hat{Q} \phi_2) \,dx = \int (\hat{Q}\phi_1)^*\phi_2 \,dx $$
Note: The equation applies for any two function because of the definition of Hermiticity (of an Operator).
Using eq.(5) - eigen value equation
$$ \int \phi_1^* (\lambda_2\phi_2) \,dx = \int (\lambda_1^* \phi_1^*) \phi_2 \,dx $$
Since $\lambda_1\, , \, \lambda_2 $ are eigen values, they are real numbers. So, $$ \lambda_1^* = \lambda_1 $$
$$ \lambda_2 \int \phi_1^* \phi_2 \,dx = \lambda_1 \int \phi_1^* \phi_2 \,dx $$
which gives that,
$$ (\lambda_2 - \lambda_1) \int \phi_1^* \phi_2 \,dx = 0 $$
We assumed that eigenvalues are distinct, so $$ \lambda_2 - \lambda_1 \neq 0 $$
The only possibility is that, $$ \int \phi_1^* \phi_2 \,dx = 0 \,\,\,\,\, \ldots... eq.(8)$$
It is possible only when the eigen functions are orthogonal to each other. Thus we get that, the eigenfunctions corresponding to distinct eigenvalues are orthogonal to each other. And it is always possible to make the eigenfunctions as the basis for our Linear vector Space (even when there are same eigen values for different eigenfunctions - it is called degeneracy. Those cases should be dealt separately).
Now, we will see our fourth postulate, that is, "The number of measurement that will result into the particular eigenvalue is proportional to the square of the magnitude of the coefficent of that particular eigenfunction in the expansion of the wave function.
When, it is said "the expansion", remember how the wave function (Ket vector) was written in terms of the basis vectors in eq.(1).
But now, we have learnt that, "Eigen functions are always the one of the possible set of basis vectors" and so, the wave function in general can always be expanded in terms of these new eigen functions as the basis (similar to eq.(1)),
$$ \vert\psi(t)\rangle = A_1(t) \vert\phi_1\rangle + A_2(t) \vert\phi_2\rangle + ...$$
Where $\vert\phi_1\rangle , \vert\phi_2\rangle $ are eigen functions and let us say their eigen values are $\lambda_1, \lambda_2 $.
Now, the postulates interprets that, the probability of a particular eigen value $\lambda_1 or \lambda_2$ is proportional to the square of the magnitude of the coefficient of the eigen function in the expansion of wave function i.e. $A_1 $ and $ A_2 $
where, $ A_1^2 $ gives the probability of the eigen value $\lambda_1 $ with the eigen function $\phi_1$ and similarly for others.
Using inner product, the value of ,
$$ A_1^2 = |\langle\phi_1\vert\psi(t)\rangle|^2 $$
In general,
$$ A_b^2 = |\langle\phi_b\vert\psi(t)\rangle|^2 \,\,\,\, \ldots...eq.(9)$$
This is the more general Statement of Max Born's interpretation.
There are many things you can predict with this postulate and it plays the most significant role hereafter.
When, I say eigen functions, there are really infinite possible sets of eigen functions you can choose in Nature. Depending on the problems, we will mostly used to deal with Energy, Position and Momentum Eigen functions.
In the beginning itself, we mentioned.. dealing with abstract ket vectors is not possible and so it should be taken the projection of these ket vectors in a desired function space.
In particular, when I take the projection on position space, we get Position Space wave function expressed as the linear combination of eigen function in position space as, from eq.(2),
$$ \langle{x}\vert\psi(t)\rangle = \sum_b A_b(t) \langle{x}\vert\phi_b\rangle \,\,\,\, \ldots...eq.(2)$$
$$ \psi(x,t) = \sum_b A_b(t) \phi_b(x) $$
where $\phi_b(x) $ is some general eigen function in position space.
But, if the eigen functions itself are position eigen functions?
Then, the inner product $\langle{x}\vert\psi(t) $ becomes dirac delta function, which means, it gives the value "one" only at that specific "x" and gives the value zero for all other x.
The reason is, when you are in one eigen state, you can only measure the corresponding eigen value and not some other eigen value of some different eigen function.
Therefore, eq.(2) becomes,
$$ \langle{x}\vert\psi(t)\rangle = \sum_b A_b(t) \langle{x}\vert{x'}\rangle = \sum_b A_b(t) \delta(x-x') $$
Thus, we obtain the value of probability amplitude,
$$ \psi(x,t) = A_b(t) $$ and $$ |\psi(x,t)|^2 = A_b^2 \,\,\,\, \ldots...eq.(10)$$
where $A_b^2 $ is the probability of getting the eigen value corresponding to eigen function $\vert{x}\rangle $.
Equating with the left side, we get the general property of a "Position space Wavefunction" and that is "the square of the amplitude of the Position space wave function gives the probability of finding the particle at x and x+dx".
This is popularly known as the Max Born Interpretation in the basic Quantum Mechanics.
Just because we found a meaning for the $ |\psi(x,t)|^2$ doesn't mean, we can only define it in terms of position eigen functions.
Still, we could expand this "Position space wave function in terms of Energy eigen functions or Momentum Eigenfunctions or anything, but in Position space (means Position as the parameter).
Instead of Position space, we can also take the inner product with momentum and get momentum space wave function and can define similar things. In that case,
$$ \langle{p}\vert\psi(t)\rangle = \psi(p,t) $$
and $$ |\psi(p,t)|^2 $$ gives the probability of finding the particle at momentum p and p+dp.
All you need to understand is the Vector space and its basis.
For a vague imagination, you can compare it with various coordinates like explaining a vector in terms of cartesian or spherical or cylindrical or etc. But, it is just for an understanding.
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