Proof:
For every basis vector

Similarly,

.

Proof:
Assume that U is unitary and that {|u_i>} is a basis and that U|u_i>=|t_i>.  Then

<t_i|t_j>=<u_i|U^TU|u_j>=<u_i|u_j>=d_ij.

The vectors {|t_i>} are orthonormal.  Are they a basis?  We have to show that every |y> can be expanded in terms of the {|t_i>}.  Let |y> be an arbitrary vector in a vector space E.

E, therefore  

. 

It follows that

, 

or 

.

Therefore {|t_i>} is a basis.

Changing a basis is therefore called a unitary transformation.

A unitary transformation is equivalent to a change of basis.

Let U be the unitary transformation transforming {|u_i>} into {|t_i>}.  The matrix elements of U in the first basis are <u_i|U|u_j>=<u_i|t_j>=U_ij.  We find the components of a vector |y> in the second basis in terms of the components in the first basis using

But

.

Therefore

 .

Similarly,

 .

Example:

The components of |A> in the {|x_i’>} basis are given in terms of the components of |A> in the {|x_i>} basis.

.

What happens to the components of a second vector |B>=W|A>?

in terms of the components in the {|x_i>} basis.

We can write

.

The matrix of W in the {|x_i’>} basis is equal to the matrix of U^TWU in the {|x_i>} basis.

Example:

, and we have

.

The matrix of W in the {|x_i’>} basis is

,

and we have

,

exactly the same as in the {|x_i>} basis.

Summary

You change basis using a unitary transformation U.  The components of each vector in the new basis are related to the components in the old basis through  .  The matrix elements of any operator in the new basis are related to the matrix elements in the old basis through .

The operator W does something different to the {|x_i’>} basis vectors then to the {|x_i>} basis vectors.  Let W’ be the operator that does to the new basis vectors what W does to the old basis vectors.

Example:

The matrix elements of W’ in the {|x_i’>} basis are the same as the matrix elements of W in the {|x_i>} basis.

W’_i’j’ =W_ij.

But

W’_i’j’ =(U^TW’U)_ij

and therefore

W_ij=(U^TW’U)_ij or (UWU^T)_ij=W_ij^’.

W’ and UWU^T have the same matrix elements in a given basis.  We therefore have W’=UWU^T.

Summary

When we change bases with the unitary transformation U, the matrix elements of every operator W change. The matrix elements of W in the new basis are equal to the matrix elements of U^TWU in the old basis.

There is an operator which has the same matrix elements in the new basis as W has in the old basis. This operator is UWU^T.

Active and passive transformations

In our example, rotating the coordinate system counterclockwise through an angle f is mathematically described in exactly the same way as rotating the vector clockwise through an angle f and leaving the coordinate system fixed.

This generalizes to :

Changing a basis from {|u_i>} to {U|u_i>=|t_i>} is mathematically exactly the same as changing every vector |y> into U^T|y>=|y’> and keeping the basis fixed.

Let  

, . 

This is the same relation between the components that we found when changing basis.

We distinguish between active and passive unitary transformations.

Active: Change all vectors |y> to U^T|y>.

Passive: Change the basis from {|u_i>} to {U|u_i>}.

Active: The operator W still does the same thing to all basis vectors.  But if |f>=W|y> then |f’>¹W|y’>.  We have |f’>=U^T|f>=U^TW|y>=U^TWU|y’>.  In our example the rotated vector has a different projection onto the x axis.

Passive: The operator  W changes into W’, which does the same thing to the new basis vectors as W does to the old basis vectors.  W’=UWU^T. In our example the projector onto the x axis becomes the projector onto the x’ axis.

Example:

 .

Project A onto the x-axis.
The projector onto the x axis is WA'=(cosqA₁+sinqA₂).

(b) Passive Transformation: Rotate the coordinate system counterclockwise.
Do not change A.  Project A onto the x'-axis.

Change W into W’=UWU^T.  The projection onto the x’ axis is UWU^TA.

In matrix notation we find the matrix elements of UWU^T and the components of A in some basis.  Let us choose the new basis.  Then .  The matrix of UWU^T in the new basis is the same as the matrix of W in the old basis.

In the new basis  .

Therefore UWU^TA=(cosqA₁+sinqA₂).  The results of the active and the passive transformation are the same.

We will soon show that the operator describing the evolution of a physical system is a unitary operator.  Therefore the evolution is a unitary transformation.

There are two ways of looking at a unitary transformation.  The Schroedinger picture of quantum mechanics treats it as an active transformation.  The state vector changes with time, the operator does not change unless it contains time explicitly.  The wave function evolves in time.  The Schroedinger equation describes this evolution in a particular representation.

The Heisenberg picture of quantum mechanics treats the unitary transformation as a passive transformation.  The state vector is fixed, the operators change with time.  Both approaches yield the same predictions.