Ket Space and Operators

First we will introduce the idea of Hilbert Space, which was named after D. Hilbert. Hilbert Space is a nondenumerable infinite complex vector space. Complex space, being a collection of complex numbers \(\mathbb{C}\) with an added structure. The infinite dimensions of Hilbert Space represents a continious spectra of alternative physical states. Alternative physical states, for example, being the position (coordinates) or momentum of a particle.

In quantum mechanics, we represent a physical state as a state vector. This state vector, again, is a part of our complex vector space. Following Dirac, this state vector is called a ket. A ket is represented by \(|\alpha\rangle\). It is postulated that a ket contains complete information about the physical state. Meaning, in theory, a ket contains everything we are allowed to know about the state state. We say "allowed to know", because of the limiting information we can know about a state due to the uncertaintiy principal. To add two kets, we can do:

\(|\alpha\rangle + |\beta\rangle = |\gamma\rangle \)

the sum of two kets is \(|\gamma\rangle\), or just another ket. To multiply a ket \(|\alpha\rangle\) by a complex number \(c\), we can do:

\(c|\alpha\rangle = |\alpha\rangle\)

where the product is another ket. Note, the complex number \(c\) can be to the left or right of the ket, it does not matter. If \(c\) is equal to zero, than the resulting ket is called a null ket.

Next, we will talk about operators. Observables, like momentum and spin, cab be represented by operators. An operator could be denoted as \(A\) for the vector space. Operators act on kets from left to right:

\(A\cdot (|\alpha\rangle) = A|\alpha\rangle\)

Our result is another ket. Most often, \(A\) is not a constant times \(|\alpha\rangle\).

Eigenkets of operator \(A\) are specific kets of importance denoted by:

\(|\alpha'\rangle,|\alpha'''\rangle,|\alpha'''\rangle,...\)

The property can be written as:

\(A|a'\rangle=a'|a'\rangle,\;\;A|a''\rangle=a''|a''\rangle\)

Values of \(a',a'',...\) are numbers. So what we are doing is taking our operator and multiplying it by the eigenket. The set of numbers that make up the eigenket \(\left\{a',a'',a''',...\right\}\) is called the set of eigenvalues of operator \(A\). The set of eigenvalues can be more compactly denoted by \(\left\{a'\right\}\).

An eigenstate is a physical state corresponding to an eigenket. To represent this idea, we will use a \(\frac{1}{2}\) systems. The relationship between eigenvalue to eigenket can be expressed as:

\(S_{z}|S_{z};+\rangle=\frac{\hbar}{2}|S_{z};+\rangle\), \(S_{z}|S_{z};-\rangle=-\frac{\hbar}{2}|S_{z};-\rangle\)

We can futher compact this to:

\(S_{z}|S_{z};\pm\rangle=\pm\frac{\hbar}{2}|S_{z};\pm\rangle\)

or

\(S_{z}|\pm S_{z}\rangle=\pm\frac{\hbar}{2}|\pm\frac{\hbar}{2}\rangle\)

Where the number of dimensions of the vector space is determined by the number of alternatives in the state of a system.

Usually, when it comes to an observable \(A\) we are talking about an \(n\)-dimensional vector space spanned by \(n\) eigenkets. An arbirtary ket \(|\alpha\rangle\) can be defined as:

\(|\alpha\rangle=\sum_{a'}^{a^{n}}c_{a'}|a'\rangle\)

where \(c_{a'}\) is a complex coefficent.

Bra Space and Inner Products

Bra space is a vector space that is "dual to" the respective ket space. Meaning, it is thought that for every ket \(|a\rangle\) there exists a bra \(\langle\alpha|\) in the dual space or bra space.

To continue.

Eigenkets as Base Kets

Given some ket \(|\alpha\rangle\) in the ket space spanned by eigenkets of \(A\), we will expand it to:

\(|\alpha\rangle=\sum_{\alpha}c_{a'}|a'\rangle\)

To find the expansion of of the coefficent we multiply \(\langle a''|\) on the left while using the orthonormality principal:

\(c_{a'}=\langle a|\alpha\rangle\)

Subsituting the exapansion of the coefficent into the some ket \(|\alpha\rangle\) we get:

\(|\alpha\rangle=\sum_{\alpha}|a'\rangle\langle a'|\alpha\rangle\)

Now, using the associative axiom of multiplication we can say that for some ket \(|\alpha\rangle\) we must have an identity operator:

\(|\alpha\rangle=\sum_{\alpha}|a'\rangle\langle a'|=1\)

This equation is known as the completeness realtion or closure.

Using this identity operator between \(\langle\alpha|\) and \(|\alpha\rangle\) we get:

\(\langle\alpha|\alpha\rangle=\langle\alpha|\cdot \left(\sum_{a'}|a'\rangle\langle a'|\right) \cdot|\alpha\rangle\)

\(=\sum_{a'}|\langle a'|\alpha\rangle|^2\)

If \(\langle\alpha|\) is normalized then:

\(\sum_{a'}|\langle a'|\alpha\rangle|^2=\sum_{a'}|c_{a'}|^2=1\)

We can also define another type of operator; the projection operator denoted by \(\Lambda_{a'}\) defined as:

\(\Lambda_{a'}\equiv|a'\rangle\langle a'|\)

The completeness relation can now be also be defined as:

\(\sum_{a'}\Lambda_{a'}=1\)

Matrix Representations

To continue.