A summary of formalism of quantum mechanics

Now we have some ideas of physical characteristics in quantum mechanics. But the real power of quantum mechanics lies in its exact mathematical formalism. It is summarized in this section.

In quantum mechanics, a system's state is represented by a vector of complex numbers and is written as $\vert a\rangle$ (called a ket vector). There is another kind of state vector called bra vector, which is denoted by $\langle \cdot \vert$. The scalar product of a bra vector $\langle b\vert$ and $\vert a\rangle$ is a linear function that is defined as follows: for any ket $\vert a' \rangle$, the following conditions are fulfilled,

$$\langle b\vert\{\vert a \rangle + \vert a' \rangle\}=\langle b \vert a \rangle + \langle b \vert a' \rangle,$$

$$\langle b\vert\{c \vert a' \rangle\}=c \langle b \vert a' \rangle,$$

$c$ being any complex number. There is a one-to-one correspondence between the bras and the kets if the conditions above are taken, with $\langle b\vert$ replaced with $\langle a\vert$, in addition to a definition that the bra corresponding to $c\vert a \rangle$ is $\bar c$ times the bra of $\vert a\rangle$. The bra $\langle a\vert$ is called the conjugate imaginary of the ket $\vert a\rangle$. Furthermore, we assume

$$\langle b\vert a \rangle = \overline {\langle a\vert b \rangle}.$$

Replacing $\langle b\vert$ with $\langle a\vert$, we find that $\langle a \vert a \rangle$ must be a real number. In addition, it is assumed

$$\langle a \vert a \rangle > 0,$$

except when $\vert a \rangle=0$. Operations can be performed on a ket $\vert a\rangle$ and transform it to another ket $\vert a' \rangle$. There are operations on kets which are called linear operators, which satisfy the following: for a linear operator $\alpha$,

$$\alpha \{\vert a \rangle + \vert a' \rangle\}=\alpha \vert a \rangle + \alpha \vert a' \rangle,$$

$$\alpha\{c \vert a \rangle\}=c \alpha \vert a \rangle,$$

with $c \in \mathbb{C}$ being a complex number. Furthermore, the sum and product of two linear operators $\alpha$ and $\beta$ are defined as follows,

$$\{\alpha + \beta \} \vert a \rangle=\alpha \vert a \rangle + \beta \vert a \rangle,$$

$$\{\alpha \beta \} \vert a \rangle= \alpha \{\beta \vert a \rangle\}.$$

Generally speaking, $\alpha \beta$ is not necessarily equal to $\beta \alpha$. Together with the definition of bra, one can define the adjoint of an operator $\alpha$ by defining that the ket corresponding to $\langle a \vert\alpha$ is $\bar \alpha \vert a \rangle$, in which $\bar \alpha$ (also denoted as $\alpha ^{\dag }$) is called the adjoint of $\alpha$. There is a special kind of operator that satisfies

$$\xi^{\dag }=\xi.$$ (3.2)

This kind of operators is called Hermitian.. They are the counterparts of real numbers in operators. In quantum mechanics, all meaningful dynamical variables in quantum physical systems are represented by Hermitian operators. More specifically, every experimental arrangement in quantum mechanics is associated with a set of operators describing the dynamical variables that can be observed. These operators are usually called observables. For an Hermitian operator (an observable) $\xi$, there is a set of kets (or states) that satisfies

$$\xi \vert x \rangle = \lambda \vert x \rangle,$$

with $\lambda \in \mathbb{R}$ and $\vert x \rangle \ne 0$. The ket $\vert x \rangle$ here is called an eigenket or eigenstate of $\xi$ and $\lambda$ is called an eigenvalue of $\xi$. Eigenvalues can be either discrete or continuous. For brevity, the discrete eigenvalues are enumerated with a subscript (e.g. $\xi_i$) and their corresponding eigenstates with norm equal to one (i.e. $\langle \xi_i \vert \xi_i \rangle=1$) are written as $\vert \xi_i \rangle$. Eigenkets that have continuous eigenvalues (e.g. $\xi '$) with norm equal to one (i.e. $\langle \xi ' \vert \xi ' \rangle=1$) are labeled with their eigenvalues. It can be shown that

$$\langle \xi_i \vert \xi_j \rangle=\delta_{ij}$$ (3.3)

where $\xi_i$ and $\xi_j$ are discrete eigenvalues and $\delta_{ij}$ is Kronecker delta function

$$\left. {\matrix{{\delta _{ij}=1\ if\ i=j}\cr {\delta _{ij}=0\ if\ i\ne j}\cr }} \right\}$$

and

$$\langle \xi ' \vert \xi '' \rangle=\delta(\xi ' - \xi '')$$ (3.4)

where $\xi '$ and $\xi ''$ are continuous eigenvalues and $\delta(\cdot)$ is the Dirac delta function

$$\left. {\matrix{{\int_{-\infty }^\infty {\delta (x)dx=1}}\cr \cr {\delta (x)=0\ for\ x\ne 0}\cr }} \right\}$$

In the experimental arrangement, any ket $\vert p \rangle$ can be expressed as

$$\vert p \rangle=\int{\vert \xi ' \rangle d\xi ' \langle \xi ... ...ts_r {\vert \xi ^{r} \rangle \langle \xi ^{r} \vert p \rangle}$$ (3.5)

where $\vert \xi ' \rangle$ and $\vert \xi ^{r} \rangle$ are all eigenkets of $\xi$. Moreover,

$$\int{\vert \xi ' \rangle d\xi ' \langle \xi ' \vert}+\sum\limits_r {\vert \xi ^{r} \rangle \langle \xi ^{r}\vert}=1$$

An abstract space in which every state can be expressed as in Equation 3.5, is called a Hilbert space. The set of $\{\vert\xi ' \rangle\}$ is called the orthonormal basis or eigenbasis of the Hilbert space. Given an eigenbasis, it is convenient to express a ket as a column vector of complex numbers whose components are the projection of the ket on the kets of the basis. This is called a representation of the ket. Specifically, a ket $\vert p \rangle$ can be represented as

$$\vert p \rangle = (\langle \xi_1 \vert p \rangle, \langle \xi_2 \vert p \rangle \cdots )^t.$$ (3.6)

where ${}^t$ denotes the transpose of a vector. The conjugate imaginary of $\vert p \rangle$ is then a row vector

$$\langle p \vert = (\langle p \vert \xi_1 \rangle, \langle p \vert \xi_2 \rangle \cdots ).$$ (3.7)

It is clear that if a ket is represented by $\vec p$, the bra corresponding to $p$ is $((\vec p) ^*)^t$ which is the conjugate transpose of the vector $\vec p$. Furthermore, for two vectors $\vec p_1$ and $\vec p_2$, $\langle p_1 \vert p_2 \rangle$ is a complex number

$$\langle p_1 \vert p_2 \rangle = (\vec p_1) ^* \cdot \vec p_2.$$ (3.8)

where $\cdot$ is the usual inner product of vectors. A linear operator $\alpha$ can be represented by a matrix

$$\left( {\matrix{{\left\langle {\xi _1} \right\vert\alpha \le... ...ght\rangle }&\cdots \cr \vdots &\vdots &\cdots \cr }} \right).$$ (3.9)

With this representation, it is clear that for an operator $\alpha$, the adjoint of $\alpha$ is

$$\alpha ^\dag =(\alpha ^*)^t.$$ (3.10)

In this thesis, only operators with discrete eigenvalues are used. Furthermore, while the dimension of a Hilbert space can be infinite, the dimensions of bases used in this thesis are finite. In this sense, a ket is a finite-dimensional vector with complex components and an operator is a matrix with complex components.

There is a class of operators that preserve the norm of kets (i.e. $\langle p' \vert p' \rangle=\langle p \vert p \rangle$ with $\vert p' \rangle=U\vert p \rangle$). These matrices are called unitary. Specifically, a unitary operator is an operator with the following property

$$U^\dag U=UU^\dag =I.$$ (3.11)

where $I$ is the identity operator (i.e. $I\vert x \rangle=\vert x \rangle$ for any $\vert x \rangle$).

The physical interpretation of Hermitian operators is the following. Given an Hermitian operator $\xi$ pertaining to a particular dynamical variable (e.g. coordinate) in a particular experimental setup, each time one makes a measurement, exactly one of the eigenket (or eigenstate) will manifest itself and the eigenvalue thereof is the measured quantity. This is sometimes called the collapse of the wave function. Recall that the eigenvalues of an Hermitian operator are real, consequently, all the physical quantities are real. Furthermore, a state in quantum mechanics describes the experiment outcomes stochastically. Specifically, if a measurement is performed on a state described in Equation 3.5, the probability of getting the outcome $\xi_i$ is

$$P(\xi_i)=\left\vert \langle \xi_i \vert p \rangle \right\vert^2$$ (3.12)

for discrete eigenvalues. For continuous eigenvalues, the probability of measuring $\xi '$ within an infinitesimal interval of $d\xi$ is

$$P(\xi ') d \xi =\left\vert \langle \xi ' \vert p \rangle \right\vert^2 d \xi.$$ (3.13)

where $P$ is usually called the probability density function (PDS). In general, for any observable $\eta$, the average value of the corresponding physical quantity is

$$\langle \eta \rangle=\langle x \vert \eta \vert x \rangle.$$

We are now ready to discuss motion in quantum mechanics, starting with an analogy between quantum mechanics and classical mechanics. In classical mechanics, any two dynamical variables $u$ and $v$ have a Poisson Bracket (P.B.), denoted by $\{u,v\}_{P.B.}$, which is defined by

$$\left\{ {u,v} \right\}_{P.B.}=\sum\limits_r {\left( {{{\parti... ...er {\partial p_r}}{{\partial v} \over {\partial q_r}}} \right)}$$

where $q_r$ and $p_r$ are canonical coordinates and momenta.

In quantum mechanics, the quantum P.B. of two operators $u$ and $v$ is defined as

$$[u,v]\equiv uv - vu =i\hbar \{u, v\}_{P.B.}$$ (3.14)

where $[u,v]$ is also called the commutator of $u$ and $v$. For canonical momenta and coordinates, it can be easily confirmed that

$$q_rq_s-q_sq_r= 0 ,$$ (3.15)

$$p_rp_s-p_sp_r= 0 ,$$ (3.16)

$$q_rp_s-p_sq_r= i\hbar \delta_{rs}.$$ (3.17)

which are the fundamental quantum conditions. These conditions also show that classical mechanics may be regarded as the limiting case of quantum mechanics when $\hbar$ tends to zero.

The variance of a physical quantity is defined as

$$\Delta \alpha \triangleq \sqrt { \langle (\alpha - \langle \alpha \rangle)^2 \rangle}.$$ (3.18)

If two observables $\alpha$ and $\beta$ do not commute (i.e. $[\alpha, \beta] \ne 0$), it can be shown by applying Schwarz's Inequality that

$$\Delta \alpha \Delta \beta \ge {1 \over 2}\left\vert {\left\langle {\left[ {\alpha,\beta} \right]} \right\rangle } \right\vert$$

where $\left[ {\alpha,\beta} \right]$ is the commutator of $\alpha$ and $\beta$. Specifically, the Heisenberg's Uncertainty Principle (Equation 3.1) can be established. Moreover, $q$'s (or $p$'s) alone form a complete set of observables on which a state in quantum mechanics can be represented. In fact, the momentum is an operator represented by coordinate $q$'s:

$$p_r=-i\hbar {\partial \over {\partial q_r}}.$$

The evolution of a closed quantum system is governed by the equation of motion. It can be written as:

$$i\hbar {\partial \over {\partial t}}\left. {\psi (t)} \right\rangle =H\left. {\psi (t)} \right\rangle,$$ (3.19)

where $H$ is the Hamiltonian (energy), being an Hermitian operator. That is:

$$H^\dag =H.$$

Equation 3.19 is known as Schrödinger's wave equation and its solutions $\psi (t)$ are time-dependent wave functions. In the literature, this is called the Schrödinger picture. In Schrödinger picture, the state of undisturbed motion is described by a moving ket with the state at time $t$ represented by $\left \vert\psi (t)\right\rangle$. The time dependent wave function $\psi (t)$ representing a stationary state of energy $H$ (associated with a Hamiltonian operator $H$) will evolve with time according to the law

$$\psi (t)=\psi _0e^{-iHt/\hbar },$$ (3.20)

where $\psi _0$ is the wave function at $t=0$. Because $H$ is Hermitian, it is clear that $e^{-iHt/\hbar }$ is a unitary operator, because according to Equation 3.11,

$$e^{-iHt/\hbar } \{e^{-iHt/\hbar }\}^\dag =\{e^{-iHt/\hbar }\}^\dag e^{-iHt/\hbar }= e^{iHt/\hbar }e^{-iHt/\hbar }=I,$$

where $I$ is the identify operator.

A quantum mechanical system is linear. That is, if $\vert s_1\rangle$ and $\vert s_2\rangle$ are both physical states allowed by a particular quantum system, a superposition of them

$$\vert s'\rangle=c_1 \vert s_1\rangle + c_2 \vert s_2\rangle$$

with $c_1,c_2 \in \mathbb{C}$ being complex numbers, is also a physical state which is allowed by the quantum system.

In this thesis, the operators are represented by matrices with finite dimensions. For an Hermitian matrix $A$, $e^{iA}$ is defined as

$$e^{iA}=\sum\limits_{n=0}^\infty {{{i^nA^n} \over {n!}}}.$$