Formal Quantum Mechanics

Postulate 1

There exists a wave function ψ from which all possible predictions about the physical properties of the system can be obtained.

The wave function is a complex, continuous, square-integrable, single-valued function of the parameters of all the particles and of time.

Born rule

The probability density for finding the particle at a specified position is $|ψ|^2$, where $|ψ(\mathbf{r})|^2dτ$ is the probability of finding the particle inside a volume element $dτ = d^3\mathbf{r}$ located at vector position $\mathbf{r}$.

Superposition principle

If $ψ_1$ and $ψ_2$ are two possible states of a given system, then the linear combination (superposition) of the two states, $ψ = c_1ψ_1 + c_2ψ_2$ is also a valid state of the system, where $c_1$ and $c_2$ are constants.

Inner product

The inner product (scalar product) between wavefunctions $ψ(\mathbf{r})$ and $ϕ(\mathbf{r})$ is defined as $(ψ(\mathbf{r}),ϕ(\mathbf{r})) =\int ψ(\mathbf{r})^*ϕ(\mathbf{r})\,dτ$ in integral form, and satisfies:

• Exchanging order of wavefunctions gives the complex conjugate,$(ϕ(\mathbf{r}),ψ(\mathbf{r})) = (ψ(\mathbf{r}),ϕ(\mathbf{r})) ^∗$

• Linearity in the second argument, $(ψ(\mathbf{r}),,αϕ_1(\mathbf{r}) + βϕ_2(\mathbf{r})) = α(ψ(\mathbf{r}),ϕ_1(\mathbf{r})) + β(ψ(\mathbf{r}),ϕ_2(\mathbf{r}))$

• Expansion of the inner product goes like $(αψ_1(\mathbf{r}) + βψ_2(\mathbf{r}),,ϕ(\mathbf{r})) = α^∗(ψ_1(\mathbf{r}),ϕ(\mathbf{r})) + β^∗(ψ_2(\mathbf{r}),ϕ(\mathbf{r}))$

• Taking the inner product of a wavefunction with itself will always be real and non-negative, (ψ(\mathbf{r}),ψ(\mathbf{r})) > 0

Postulate 2

To each observable quantity is associated a linear, Hermitian operator.

The eigenvalues of the operator represent the possible results of carrying out a measurement of the corresponding quantity.

Immediately after making a measurement, the wavefunction is identical to an eigenfunction of the operator corresponding to the eigenvalue just obtained as the measurement result.

Linear operator

An operator is linear if $\hat A(c_1ψ_1+c_2ψ_2)=c_1 \hat Aψ_1+c_2 \hat Aψ_2$

Hermitian operator

A linear operator $\hat A$ is Hermitian if $\hat A= \hat A^†$ , where the Hermitian conjugate is defined by $\int\psi^*_1\left(\hat A\psi_2\right),\mathrm{d}\tau=\int\left(\hat A^†\psi_1\right)^*\psi_2,\mathrm{d}\tau$ for any functions.

This can also be written $\left(\psi_1,,\hat A\psi_2\right)=\left(\hat A^†\psi_1,,\psi_2\right)$

Properties of Hermitian operators

• The eigenvalues are real quantities

• The eigenfunctions $\phi_n,\,\phi_m$ corresponding to different eigenvalues are orthogonal, i.e., $\int\phi_n^*(\mathbf{r})\phi_m(\mathbf{r})\,\mathrm{d}^3r=0$

Degenerate eigenvalue

An eigenvalue is degenerate if more than one eigenvector is associated with it, i.e., more than one eigenfunction for one eigenvalue.

A set of eigenfunctions can always be chosen to be orthogonal for degenerate eigenvalues.

Postulate 3

The operators representing the position and momentum of a particle are $\hat R=\mathbf{r}$ and $\hat P=-i\hslash\nabla$ in three dimensions.

Operators representing other dynamical quantities bear the same functional relation to these as do the corresponding classical quantities to the classical position and momentum variables.

Expansion postulate

The expansion postulate tells us that that for every dynamical observable that can be measured, its eigenfunctions form a complete set.

Written $\psi=\sum_nc_n\phi_n$ where the constants $c_n$ are the expansion coefficients, given by $c_n=\int\phi_n^*\psi\,\mathrm{d}\tau=(\phi_k,\psi)$.

Basis

A complete set of functions that are linearly independent.

Orthogonal wavefunctions are always linearly independent, so a complete set of orthonormal wavefunctions can also be called an orthonormal basis.

Postulate 4

When a measurement of a physical variable represented by a Hermitian operator $\hat A$ is carried out on a system whose wavefunction is $\psi$, then the eigenvalues of $\hat A$ represent the possible outcomes of the measurement, and the probability of the result being equal to a particular eigenvalue $\lambda_m$ is $|c_m|^2$ where, $\psi(\mathbf{r})=\sum_nc_n\phi_n(\mathbf{r})$ and $\phi_n(\mathbf{r})$ are orthonormal eigenfunctions of $\hat A$ corresponding to eigenvalues $\lambda_n$. This probability is found using the formula given by the expansion postulate.

Expectation value

For an identical experiment repeated many times, always with the system in the same initial state ψ, the average value of A is called the expectation value, $\langle A\rangle=\int\psi^*\hat A\psi,\mathrm{d}\tau=(\psi,\hat A\psi)$

Postulate 5

Between measurements, the evolution of the wave function of an isolated system with time is governed by the Time-Dependent Schrödinger equation, $\hat H \Psi(\mathbf{r},t) =i\hslash\frac{\partial \Psi(\mathbf{r},t) }{\partial t}$ where $\hat H$ is the Hamiltonian operator for the system.

If the potential is independent of time, the TDSE separates with solutions of the form $\phi(x)e^{-iEt/\hslash}$, and the general solution (by the expansion postulate) is $\psi(x, t)=\sum_nc_n\phi_n(x)e^{-iE_nt/\hslash}$ in one dimension, where $\phi_n(x)$ satisfy the one-dimensional TISE with eigenvalues $E_n$.

Inner product in Dirac notation

The inner product is written as $⟨ψ_1|ψ_2⟩ ≡ (|ψ_1⟩,|ψ_2⟩)$.

• In exchanging the order of the wavefunctions, we write $⟨ψ_1|ψ_2⟩ = ⟨ψ_2|ψ_1⟩^∗$

• A state vector $|ψ_1⟩$ is normalised if $⟨ψ_1|ψ_1⟩ = 1$ and two states are orthogonal if $⟨ψ_1|ψ_2⟩ = 0$

• Therefore, the condition for a set of states ${|ϕ_j⟩}$ to be orthonormal is $⟨ψ_j|ψ_k⟩ = δ_{j,k}$

Inner product in column vector representation

For a pair of complex column vectors, $\mathbf{c}, \,\mathbf{d}$, the inner product is defined as $(\mathbf{c},\mathbf{d})= \sum_n c^∗ _nd_n=(\mathbf{c}^{∗})^T\mathbf{d}$.

Superposition of a set of orthonormal states in Dirac notation

$|ψ⟩= \sum_n c_n|ϕ_n⟩$ and the corresponding bra vector is $\langle\psi|=\sum_nc_n^*\langle\phi_n|$.

Representation of a ket in column notation

$$|\psi\rangle=\sum_jc_j|\phi_j\rangle\qquad\text{can be represented as}\qquad \mathbf{c}=\begin{bmatrix}c_1\\c_2 \\ c_3\\vdots\end{bmatrix}$$

Representation of a bra in column notation

\langle\psi|=\sum_jc_j^*\langle\phi_j|\qquad \text{can be represented as}\qquad(\mathbf{c}^)^T=\begin{bmatrix}c_1^* &c_2^* & c_3^*&\dots\end{bmatrix}

Orthogonal matrix

A matrix is orthogonal if its inverse is its transpose, $M^{-1}=M^T$.

Hermitian matrix

A matrix is Hermitian if $M=M^\dagger$, where the Hermitian conjugate is given by $M^\dagger=(M^T)^*$.

Unitary matrix

A matrix is unitary if it obeys $U^{-1}=U^\dagger$.

Outer product

Written $|\psi_1\rangle\langle\psi_2|$, and is a linear operator.

Resolution of the identity (the closure relation)

The identity operator can be written using the bras and kets of any orthonormal complete set $\{|ϕ_j⟩\}$ (orthonormal basis) as $\hat I= \sum_j |ϕ_j⟩⟨ϕ_j|$.

Matrix element of an operator

$A_{j,k}=⟨ϕ_j|\hat A |ϕ_k⟩$ is the matrix element of $\hat A$, and is a scalar.

Any operator can therefore be expressed as a sum of matrix elements and outer products between vectors from an orthonormal basis.

Hermitian conjugate in Dirac notation

The Hermitian conjugate of a ket is defined as the corresponding bra, and vice-versa, i.e., $|ψ⟩^† = ⟨ψ|$ and $⟨ψ|^† = |ψ⟩$.

The Hermitian conjugate is defined by $\left(⟨ψ_1| \hat A|ψ_2⟩\right)^* =⟨ψ_2| \hat A^†|ψ_1⟩$.

Matrix elements of $\hat A^\dagger$

The matrix elements of $\hat A^†$ are found by $(A^†)_{j,k} = ⟨ϕ_j| \hat A^†|ϕ_k⟩ = \left(⟨ϕ_k| \hat A|ϕ_j⟩ \right)^∗ = A^∗_{ k,j}$.

Change of basis

Changing basis from $|\psi\rangle=\sum_kc_k|\phi_k\rangle$ to $\sum_jd_j|\chi_j\rangle$ results in $d_j=\sum_{k}\langle\chi_j|\phi_k\rangle c_k$ where $S_{j,k}=\langle\chi_j|\phi_k\rangle$ are the matrix elements of $S$, the similarity transformation.

Similarity transformation for an operator

Switching from the representation of $\hat A$ in an old basis to $\hat B$ in a new basis is done via $B=SAS^\dagger$.

Eigenvalues relation to representations

The eigenvalues of a representation of an observable $\hat A$ are independent of the representation chosen.

Dirac delta orthonormality rule

$\langle x'|x''\rangle=\delta(x'-x'')$ where $\delta(x'-x''$) is the Dirac delta function. This replaces the discrete orthonormality rule.