Geometrical Interpretation of State Vector

The set of vectors e = {ef}"' is an orthonormal basis in the n-dimensional space, П&" when the orthonormality can indeed be expressed as the inner or dot product

In this case we say, such a set consists of n linearly independent vectors and defines an orthonormal basis in the n -dimensional space, K".

So, if e = je,}" is an orthonormal basis in the л-dimensional space П&" , then an arbitrary vector a with an inner product can be expressed uniquely in the given orthonormal basis as:

Here, a, the components of the transformed vector in the basis e, can be obtained as:

This implies the components of the given vector in a given orthonormal basis can be obtained by taking the inner product of the vector with the appropriate basis vectors. From here, it is obvious that the vector a can be expanded as:

This shows that the bases can be changed to make the problem simpler. We observe that relations 1.6 and 1.3 are the generalization of relations 1.45 and 1.46 respectively. The quantity P,* should also have the sense of the projection operator. Operators have to be transformed also, under similar transformation as in relation 1.48.

So, to conclude, any such set of say n linearly independent vectors is called a basis for the vector space and said to span the vector space. The coefficients {a,} for a particular vector a are called the components of a. Equations 1.46 or 1.48 is said to be the (linear) expansion of a in terms of the basis {e,} and the vector space is said to be the space spanned by the basis. We observe from 1.47 that an inner or dot product space is defined such that component values can only be real numbers.

From the aforementioned, the probability amplitude (Akii„), can be examined as the scalar product of the vector |//„) (state vector) on the basis vector |л*). Generally, in quantum mechanics we write in compact form general relations via state vector description without making reference to a particular representation. So, relations are described by symbols that mimic vector quantities in vector algebra as well as vector analysis and the concrete description of these relations can only be unveiled by using a defined representation.

 
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