# 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:

where,

We find the inverse transform of 1.50:

**И)**

Comparing expression U_{w} 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, P_{u}„ 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 P_{rt}, 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 P_{f},„ 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„).*

Similarly,

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 *A _{n}* of a discrete spectrum resulting in the

**closure relation:**

**2**