29/08/2023
space:
The formalism of quantum mechanics deals with
operators that are linear and wave functions that belong to an abstract Hilbert space. The math-ematical properties and structure of Hilbert spaces are essential for a proper understanding of
the formalism of quantum mechanics.
A Hilbert space consists of a set of vectors and a set of scalars which
satisfy specific properties:
One of the properties is:
Hilbert space has a defined scalar product that is strictly positive.
Scalar product is not commutative.
The scalar product is a complex number.
When the functions multiplied in opposite order then the result of the product becomes its complex conjugate.
Physical meaning of scalar product in Hilbert space:
The scalar product can be interpreted in two ways.
First, by analogy with the scalar product of ordinary vectors in the Euclidean space, where Skalar product of A and B represents the projection of vector B on vector A.
The product in of wave functions in Hilbert space also represents the projection of second function on the first one.
Second, in the case of
normalized states and according to Born’s probabilistic interpretation, the quantity of scalar product of PHI to PSI represents the probability amplitude that the system’s state PHI will, after a measurement is
performed on the system, be found to be in another state PSI.
Remember, when any operation is done on the quantum system its state is affected, doing measurements makes changes on the quantum states
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