Formalism-of-quantum-mechanics

Dirac notation or bra-ket notation is a way of writing quantum

• bra $⟨\psi |$ = linear function of a vector
• ket $|\psi ⟩$ = vector

${\psi }_1$ and ${\psi }_2$ are orthogonal $⟨{\psi }_1|{\psi }_2⟩=0$ ${\psi }_1$ and ${\psi }_2$ are normalized $⟨\psi |\psi ⟩=1$ ${\psi }_1$ and ${\psi }_2$ are orthonormal (both orthogonal and normalized)

$$\int dr{\psi }_1^{\ast }(x){\psi }_2(x)=0$$

Kronecker delta shows orthonormality $\int dr{\psi }_1^{\ast }(x){\psi }_2(x)={\delta }_{nm}=⟨n|m⟩$where

• when n $=$ m , $\delta =1$
• when n $\ne$ m, $\delta =0$

Projection = the coefficient $c_n$ is the projection of $\psi$ onto the vector $|n⟩$

$$\psi (r)=\sum _n{c}_n{\psi }_n(r)\text{ or }|\psi ⟩=c_n|n⟩$$

Dirac delta function = $\delta (r-r^{\prime })$ or ${\sum }_n|n⟩⟨n|=\delta (r-r^{\prime })$

closure relation An orthonormal basis set $\{|n⟩\}$ is a basis set if for every function $\psi (r)$, the function can be expressed in

$$\sum _n{\psi }_n^{\ast }(r^{\prime }){\psi }_n(r)=\delta (r-r^{\prime })$$

sifting property of the Dirac delta function

$$\int d\beta c(\beta )\delta (\alpha -\beta )=c(\alpha )$$

Inner product in Position representation =

$$⟨{\psi }_1|{\psi }_2⟩=\int dr{\psi }_1^{\ast }(x){\psi }_2(x)$$

in momentum representation =

$$⟨{\psi }_1|{\psi }_2⟩=\int dp\overline{{\psi }_1^{\ast }(p)}\overline{{\psi }_2(p)}$$

Hilbert space

Hilbert space is similar to a vector space

• Hilbert space is linear
• inner product exists

$$⟨{\psi }_1|{\psi }_2⟩=\int dr{\psi }_1^{\ast }(x){\psi }_2(x)$$

• length of vector = $⟨\psi |\psi ⟩$

The state of a quantum system is characterized by its state vector $|\alpha ⟩$ which is an element of the Hilbert space. All physical information about the given system is in its state vector

To say that a quantum system characterized by an n-dimensional Hilbert state means that each possible state of the system can be represented by a state vector $|\alpha ⟩$ with n complex components, and can be written as

$$|\alpha ⟩=\left[\begin{array}{c}{a}_1\\ a_2\\ .\\ .\\ a_n\end{array}\right]$$

The n-dimensional Hilbert space will have n basis, so that the state vector can be represented as:

$$⟨\alpha ⟩=a_1|{\alpha }_1⟩+a_2|{\alpha }_2⟩+...+a_n|{\alpha }_n⟩$$

While the state vector itself is basis-independent, the values of its components $\{a_i\}$ will depend on the choice of basis.

example An electron spin (up and down) system is a two-dimensional Hilbert space with the two basis being “up” and “down”, so that a general element in the Hilbert space can be represented as $|\alpha ⟩=c_{+}|+⟩+c_{-}|-⟩$

basis sets

Discrete orthonormal basis set orthogonal =

$$\int dr{\psi }_1^{\ast }(r){\psi }_2(r)={\delta }_{nm}\text{ or }⟨n|m⟩={\delta }_{nm}$$

spans the space =

$$\sum _n{\psi }_n^{\ast }(r^{\prime }){\psi }_n(r)=\delta (r-r^{\prime })\text{ or }\sum _n|n⟩⟨n|=I$$

Continuous orthonormal basis set orthogonal =

$$\int dr{\omega }_{\alpha }^{\ast }(r){\psi }_{\beta }(x)=\delta (\alpha -\beta )\text{ or }⟨\alpha |\beta ⟩=\delta (\alpha -\beta )$$

spans the space =

$$\int d\alpha {\psi }_{\alpha }^{\ast }(r^{\prime }){\psi }_{\beta }(r)=\delta (r-r^{\prime })\text{ or }\int d\alpha |\alpha ⟩⟨\alpha |=I$$

generalization of continuous basis set the wave function can be expressed as

$$\psi (r)=\int d\alpha c(\alpha ){\omega }_{\alpha }(r)$$

where $c(\alpha )=\int dr{\omega }_{\alpha }^{\ast }(r)\psi (r)$

In dirac notation =

$$|\psi ⟩=\int d\alpha c(\alpha )|\omega ⟩$$

where $c(\alpha )=⟨\alpha |\omega ⟩$

Operators

state vectors are modified by linear operators that act upon them and that determine their physical properties

observables = a mathematical transformation between two elements of a given Hilbert space and are represented by operators

The expectation value associated to a measurement of the physical observable given a quantum state is:

$$⟨O⟩=⟨\psi |\hat{O}\psi ⟩$$

Given that the outcome of any measurement is a real quantity, the expectation value is real for any operator in any given quantum state. In other words, operators representing physical observables must satisfy

$$⟨O⟩=<O{>}^{\ast }$$ $$⟨\psi |\hat{O}\psi ⟩=⟨\hat{O}\psi |\psi ⟩$$

these operators are called Hermitian operators which are by definition equal to their conjugate

• Hermitian operators have associated real eigenvalues
• The eigenvectors $|{\psi }_1⟩$ and $|{\psi }_2⟩$ associated to different eigenvalues (${\lambda }_1\ne {\lambda }_2$) are orthogonal
• The eigenvectors of a Hermitian operator span the complete Hilbert space and so represent a complete basis in the Hilbert space

The Heisenberg uncertainty principle is the consequence of the axiom that all physical observables in quantum physics are represented by Hermitian operators

Eigenvalue equations

Eigenvalue equations =  $A\vec{v}=\lambda \vec{v}$ where

• $A$ represents a square matrix of dimensions $n\times n$
• $\vec{v}$ is a column bector with dimensions n
• $\lambda$ is the eigenvalue of the equation and $\vec{v}$ is the eigenvector

To find the eigenvalues =

1. Find the characteristic equation of the matrix A

$$Det(A-\lambda \cdot I)=0$$

Where $I$ is the identity matrix and det is the determinant. This gives the eigenvalues $\lambda$ 2. Find the eigenvectors by solving $A{v}_i=\lambda v_i$

Position representation

$${\Omega }_{r^{\prime }}(r^{\prime })=\delta (r-r^{\prime })=|r^{\prime }⟩$$

Position space  is the set of all position vectors r in Euclidean space, and has the dimensions of length, where a position vector defines a point in space.

The wave function is represented as $⟨r|\psi ⟩=\psi (r)$

Position operator ($X$)

$$⟨r|X|\psi ⟩=x⟨r|\psi ⟩\text{ or }X\psi (r)=x\psi (r)$$

The position operator $X$ has the position $x$ of a quantum mechanical particles as eigenvalues

expectation value of the position components of a particle =

$$⟨\psi |X|\psi ⟩=⟨X⟩=\int dr{\psi }^{\ast }(r)x\psi (r)=\int drx|\psi (r){|}^2$$

momentum representation

$${\mu }_{p^{\prime }}(r)=(2\pi \hslash {)}^{-3/2}{e}^{\frac{i}{\hslash }{p}^{\prime }\cdot r}=|p^{\prime }⟩$$

wave function in momentum representation =

$$⟨p|\psi ⟩=\frac{1}{(2\pi \hslash {)}^{3/2}}\int dr{e}^{\frac{i}{\hslash }p\cdot r}\psi (r)=\bar{\psi }(p)$$

inner product in momentum representation =

$$⟨{\psi }_1|{\psi }_2⟩=\int dp{\bar{\psi }}_1^{\ast }(p)\overline{{\psi }_2}(p)$$

momentum operator $\hat{P}$

• in the momentum representation =

$$⟨p|\hat{P}|\psi ⟩=\mathbf{p}⟨p|\psi ⟩$$

• in position representation =

$$⟨r|\hat{P}|\psi ⟩=\frac{\bar{h}}{i}\nabla \psi (r)=\frac{\bar{h}}{i}\frac{\partial \psi (r)}{\partial x}$$

expectation value

$$<P_x>=\int dr\psi (r)\frac{\bar{h}}{i}\frac{\partial \psi (r)}{\partial x}$$ $$<P>=m\frac{\partial <x>}{\partial t}$$