n-dimensional Real Vector Space:
n-dimensional Complex Vector Space:

Vectors in a given vector space are represented using Dirac braket notation:
Where are vector that form a basis for , and are scalars.

Inner product is a map between two vectors and onto the scalars, denoted by:
Where is the field of the vector space (either or ), which satisfies the following axioms:

Inner product space is a vector space with an inner product

If one tries to distribute the ket term on the bra term:
The scalars are conjugated (anti-symmetry on the first term)

Orthogonal vectors:
Norm of a vector:
Orthonormal Basis: All basis vectors have norm 1 and orthogonal between them.

Given and the inner product must obey:

If the basis chosen was orthonormal, then and so:

The dual space of a vector space is denoted is the set of all linear functions from to the scalars.

If the elements of are column vectors, then elements of should map column vectors to scalars, therefore the elements of are row vectors.

There exists a canonical map between and such that if a vector is mapped to a co-vector denoted , the application of to a vector equals the inner product of and , denoted .

In the context of QM, vectors are called ket and co-vectors are called bra.

Given and , then for the above to be true:

Here, the vector is expanded in an orthonormal basis , where the components are , i.e. the inner product of each basis vector with .

If the ket corresponds to , then because of the conjugation applied corresponds to .

In braket notation it is allowed to place scalars inside the braket, signifying for kets:
and for bras:
More generally, an equation among kets:
This implies an adjoint equation:

The expansion
can also be "adjointed" into:
Notice how the inner product is commuted to , because in the sum that term corresponds to scalars, it has to be conjugated, which is equivalent to commuting the order of the terms.

This process converts any basis to an orthonormal one.
Non-orthonormal basis: , , etc.
Transforms to orthonormal basis: , , etc.

Schwarz Inequality:

Triangle Inequality:

Subspace: A subset of a vector space that also forms a vector space. A subspace "" of dimensionality will be denoted

Sum of subspaces: Given two subspaces, their sum can be defined as the vector space containing all elements of both vector spaces and all linear combinations.

An operator is a linear function .

The action of on can be represented as:
Operators can also act on bras:
Linear operators obey the following rues:

And similarly, for bras:

Identity operator ( ): and

Rotation operator ( ): Rotates vector by about the axis parallel to the unit vector .

The action of an operator on a basis: can be used to determine the effect on any vector:

In general, The commutator of and is defined as:

Identities:

The inverse of denoted satisfies

Distributing the inverse over a series of operations reverses the order:

When acts on the basis we obtain the new basis as done in Effects on basis, which we know the j-th component for each new basis vector in the original basis is: which is equal to:
are the matrix elements of in that basis.

If acts on to give then the components of may be calculated:
Which corresponds to the following matrix operation:

The columns correspond to the components of the transformed basis vectors.

Similarly, if acts on to give the components may be calculated as:

The projection operator for the ket is defined as:
It operates on a ket , and "projects" it onto the ket .

If one projects onto each basis vector, and then adds up all projections, then you end up with again.

Similarly, with bras:

Using the definition of the elements of a matrix representation of an operator, we obtain:
This corresponds to the matrix with zero in all elements except the i-th element in its diagonal.

The matrix representation of a product of operations is the product of the matrices of each factor:

Similarly to how a scalar operating on a vector is conjugated when the adjoint is calculated:
A vector being operated by omega can be adjointed by taking the hermitian conjugate or adjoint of the operator denoted .
Given a basis, the components of the adjoint may be calculated as:

The matrix representing is the transpose conjugate of the matrix representing .

Distribution of adjoint on operators reverses order:

When a product involving operators, bras, kets and scalars, one must reverse the order of all factors and make the substitutions , and .
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An operator is hermitian if
An operator is anti-hermitian if

The same way numbers can be decomposed into real and imaginary parts by
operators can be decomposed into hermitian and anti-hermitian parts:

An operator is unitary if this automatically implies

The product of two unitary operators is also unitary. Additionally, they are associative and always have an inverse. Therefore, they form a group, named unitary group U(n). They are the generalization of rotation in to

They preserve the inner product:

The columns of an unitary matrix are column vectors which are orthonormal to each other.

An active transformation is a linear map that maps vectors from the vector space to itself.

A passive transformation instead, is applied to operators and is applied to coordinates instead.

Eigenvalue equation:
Where is the eigenket of and its eigenvalue.

Because of the linearity of the operator, if is an eigenket, then so is .
We will not treat them as different vectors for the eigenvalue problem.

For the identity operator :

For the projection operator on a vector :

For the operator :

From we obtain:
Assuming the inverse of ) exists:
But there is no linear operator that transforms the vector to . Therefore the inverse doesn't exist, and so:
Using the Leibniz formula for determinants we can conclude that this is a polynomial equation of variable with coefficients determined by :
is the characteristic polynomial and the above equation is the characteristic equation. Its roots are the eigenvalues, independent of the basis.

Eigenvectors may be labeled by their eigenvalue. For example for :

This notation assumes that the eigenvalues are different for every vector, which only occurs if the polynomial doesn't have repeated roots (i.e. non-degenerate)

The eigenvalues of a Hermitian operator are real:
If is hermitian then is real.

For every hermitian operator there exists a basis consisting of its eigenvectors which are orthonormal. The matrix representation of the operator consists of its eigenvalues in its diagonal

If the characteristic equation of is non-degenerate then there is only one orthonormal basis formed by its eigenvectors. If it's degenerate, then there are many such basis. Eigenvectors may span an eigenspace with value .

If a root is repeated times for some eigenvalue , then there will be an eigenspace .

For degenerate eigenvalue problems, a ket in a degenerate eigenspace is denoted where labels the specific ket.

The eigenvalues of a unitary operator are complex values with unit modulus

The eigenvectors of a unitary operator are mutually orthogonal. (Assuming no degeneracy)

If starting from an orthonormal basis, one changes to the basis generated by the eigenvectors of a hermitian operator , because the eigenvectors are also orthonormal, the change of basis corresponds to a rotation of one orthonormal basis to another. This action is performed by unitary operator:

We can conclude then, that hermitian matrices (columns contains orthonormal basis), may be "rotated backwards" to the original orthonormal basis , therefore diagonalizing the matrix. In other words:
Every hermitian matrix may be diagonalized by a unitary change of basis.

Or in terms of a passive transformation:

If is a hermitian matrix, there exists a unitary matrix (Built out of the eigenvectors of ) such that is diagonal.

This is equivalent to the eigenvalue problem. The general case of matrix diagonalization is but if is hermitian, then is unitary, and .

Theorem: If and are two commuting () Hermitian operators, there exists (at least) a basis of common eigenvectors that diagonalizes them both.

If at least one of the operators are non-degenerate: Every eigenvector of is an eigenvector of and vice-versa.

Initial conditions (, , , )

This problem can be expressed as the following matrix equation:

One must think of the state of the system as vector that's a function in time: this is an abstract vector that represents the position of the system.

The most intuitive basis is the one where the unit vectors correspond to unit displacement of moth masses. This basis is denoted by and :
One may also double-differentiate the equation to obtain its second derivative:
The vectors and are related by an operator , in this basis the operator can be found to be represented by the matrix:

From now on will refer to the operator AND the matrix in this original basis.

From Newtonian mechanics, one can derive:

And in general, the following is true:

The difficulty in solving this is the coupling between two differential equations.

The trick is to change basis to one where is diagonal. This new basis will have basis vectors and . The change of basis matrix will be named .

Because is hermitian, will be unitary, and so we can write:
We will label the values of the diagonal of as:
We can easily find what happens if we feed the basis vectors and (in this new basis):
We can then write the vector form of the above equation, as its true for all basis:
From this we conclude that are the eigenvalues of and are the eigenvectors. And because of the definition of , the i-th column of U corresponds to the components of in the , basis.

In this basis, the equation becomes:

Which is much simpler to solve.

Finding the eigenvalues of can be done by the determinant method:

Solving this equation, we obtain:
And finding the eigenvectors in the original basis , we can express :

The inital conditions () in the original basis are given by To find the initial condition in the new basis, we multiply the components by to obtain:

The above matrix equation reduces to:
The above are second order homogeneous differential equations, with the following solutions for the given initial conditions:
Now we know the vector for the new basis, we can transform to the old basis by multiplying by
Substituting, we obtain: