Coordinate Transformation

Moving to another system of coordinates, equation 1.46 can be transformed:

Comparing 1.46 with 1.49 then we have the transforming of the projection a', onto a, under the action of the unitary operator U:


We find the inverse transform of 1.50:


Comparing expression Uw with U /*f we find that

So, an operator is unitary if its inverse equal to its adjoints and equation 1.53 confirms the unitarity of the matrix U . From equation 1.52, the unitary transformation is seen as a linear transformation transforming a vector to another vector in the same space. Tire projections a, and a', may be examined as the geometrical interpretation of the amplitudes a) and {/r„| a). We examine 1.50 as the geometrical interpretation of 1.14. It is instructive to note that a unitary transformation does not change the physics of a system but merely transforms one description of the system to another physically equivalent description.

Projection Operator

We examine the measurement process as projecting ket vector |a) onto the ket |//„) where the projection operator on the subspace spanned by the single orthonormal basis vector |//„):

Geometrically, Pu„ can be considered as an orthogonal projection operator onto to the ket vector |/v„) confirmed by the following properties:

Since, the ket vector |/r„) is normalized to unity: then

So, projecting twice in succession onto a given vector is equivalent to projecting once where operators with such properties are called idempotent operators. Relation 1.57 does not imply Prt, is its own inverse. It is instructive to note that projection operators are in general noninvertible.

In equation 1.54, {|/r„>} are members of an orthonormal basis. The projection operator P,,„ satisfies the Dirac formula:

This is also known as a closure relation. This formula may be generalized:

Relation 1.58 is for the discrete spectrum. For a mixed spectrum:

For any measurement, the state changes: with a probability

The projection operator Pf,„ acting on an arbitrary ket vector |я) yields a ket proportional to |/r„) with the coefficient of proportionality being (//„| a) that is the scalar product of |a) by pi„).


So, the geometrical interpretation of P,,„ is the orthogonal projection operator onto the vector |//„). It is instructive to note that P,,„ is specified when the basis |//„) is specified. So, the meaning of P,,„ will always depend on context.

Continuous Spectrum

So far, we have dealt with the discrete basis and now we examine a continuum basis. We consider the probability amplitude (£| //„) that is a function of the continuous variable £ and /r„ corresponding to a discrete spectrum. The quantity |(£|/r„)|2 for given //„ is the probability amplitude and

is the probability in the state of the finding the quantity £ within the interval £ and £ + d£. The orthonormalization condition is written:

while the superposition principle in 1.14 is now written:

and viewed as the expansion of a component of |£) in one basis, |a), into those of another basis, |//). From here we have

Considering the normalization condition 1.64 and integrating over £ then Considering that

Then for 1.67 to be equal to the LHS (Left-Hand_Side) when in the continuum basis we have

Here, <5(/r - //') is called the Dirac d-function. A similar result can be obtained for a parameter An of a discrete spectrum resulting in the closure relation:


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