Hello!
"EIGENVALUES AND EIGENVECTORS IN THE HILBERT SPACE"
For a given "T" operator in the Hilbert space, we have certain eigenvectors which mantain its direction under the "T" action
The eigenvalue is the scalar
These are the eigenvectors and the proportionalily constants are the eigenvalues
In quantum mechanics, the only one mesurable quantities are represented by operators
If a measurement is made, the system will remain in the eigenvector state related with the measured quantity (eigenvalue)
That is the reason why operators, eigenvectors and eigenvalues are so important in quantum mechanics
They are closely related with what we will be able to measure in a quantum system
An example, suppose that we have a quantum mechanical operator "A"
An we have also a ket "psi"
We draw the basis
If "A" is applied to the ket, we obtain another different ket
The new obtained ket has been rotated
It is changed
The original ket has been totally modified
Now suppose that the same operator acts on a ket, but now the modified ket holds its original direction
We draw the three dimensional axis
The eigenvectors are shown
It is also shown "psi ' "
Such a ket is an eigenvector of the operator "A"
They are proportional
The eigenvectors will define an specific geometry that will be very important for us
The eigenvectors will tell us the posible measurable states for a quantum system
We draw the axis
The eigenvector is shown
The eigenvectors will also define a new ket basis in the Hilbert space
The operator will have a diagonal form in the eigenvector basis
To put an example to help to understand it. Imagine that we have an XYZ reference system
In our space we have a cube and we want to describe it
If we put new axis on a cube vertice, the system will be very easy to describe
Just write three simple inequalities
The change of the axis position has simplified the description of the object
We can expect the same in the Hilbert space case
