• #Lecture 5
  • #Idea of cauchy sequence :-
  • #Linear operators
  • #Postulates
• #Lecture 6
  • #classical Physics / Quantum Physics
  • #Lec 6 (contd.) -- Why we need it in the relation ?
  • #The Shrodinger Equation :-
  • #statement :

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\(l_2\), \(L_2^{(a,b)}\) both are infinite diamensional LVS.

let us assume we have set of complex no.

\(\{z_n\}\), then for convergence of sequence.

\[ |z_n - z_m| \longrightarrow 0 \qquad \text{as, } n, m \to \infty \]

[\(\{z_1, z_2, z_3 \cdots z_n \cdots\}\) formed according to some rule.]

This implies

also \(\left(\Rightarrow \lim_{n\to\infty} z_n \text{ exists}\right)\) \(\lim_{n \to \infty} z_n = \) some complex no.

If we include the limit point of sequence \(z_n (n \to \infty)\), then \(\{z_1, z_2 \cdots z_n\}\) is called as closed set. (or closed sequence).

\(\Rightarrow\) \(\{|\psi_n\rangle\}\) is a cauchy sequence if \(\left(\lim_{n,m\to\infty}\big\|\psi_n - \psi_m\big\| \longrightarrow 0\right)\)

\(\Rightarrow\)

\[ \lim_{n\to\infty} |\psi_n\rangle = |\psi\rangle \]

limiting vector

if \(|\psi\rangle \in V\) then it is called complete vector space

i.e. If the limit vector of every cauchy sequence in a LVS \((V)\) is also in \(V\), then \(V\) is a complete LVS.

A complete LVS with an inner product is a Hilbert space.

\(\ast\) The reason we need such sequence is to make sure that operations with infinite diamensional vectors do not lead to errors.

\(\ast\) A Hilbert space with a denumerable basis \(\{|\phi_n\rangle\}\) (\(n = 1,2,3 \cdots\) inf) [\(\to\) (able to count)] is called a separable Hilbert space.

[\(\hookrightarrow\) we will be working in separable hilbert space in whole Q.M]

\[ \Rightarrow \quad |\psi\rangle = \sum_{n=1}^{\infty} c_n|\phi_n\rangle \] \[ c_n = \langle\phi_n|\psi\rangle \qquad \text{if } \{|\phi_n\rangle\} \text{ forms orthonormal basis.} \]

(assuming basis are orthonormal)

\[ \langle\phi_n|\phi_m\rangle = \delta_{mn} \] \[ \langle\psi|\psi\rangle = \sum_{n=1}^{\infty}|c_n|^2 = \|\psi\|^2 < \infty \qquad \text{(finite)} \in l_2. \] \[ \left\{ \begin{aligned} \sum_{n=1}^{\infty} |\phi_n\rangle\langle\phi_n| &= \mathbb{1}\\ \langle\phi_n|\phi_m\rangle &= \delta_{mn} \end{aligned} \right. \]

operators acts on \(|\psi\rangle\), where linear operator acts linearly

\[ \left\{ \begin{aligned} \hat A\big(|\psi\rangle + |\chi\rangle\big) &= A|\psi\rangle + A|\chi\rangle\\ \hat A\big(c|\psi\rangle\big) &= c\,\hat A|\psi\rangle\\ (\hat A + \hat B)|\psi\rangle &= \hat A|\psi\rangle + \hat B|\psi\rangle\\ \hat A\big(\hat B|\psi\rangle\big) &= (AB)|\psi\rangle \end{aligned} \right. \qquad \left(A, B \text{ are linear operators}\right) \]

in general \(AB \neq BA\) operators do not commute.

(1) \(\rightarrow\) Every dynamical system is described by a "state vector" \(|\psi\rangle\) which is element of a Hilbert space.

where system can be

-- 1 particle

-- 2 particle

-- 10 particle

-- Human being \(\cdots\) galaxy

\(\rightarrow\) every one of such system is described by their "own" state vector.

\(\rightarrow\) Every information is burried in this vector \(\underline{|\psi\rangle}\).

(2) \(\rightarrow\) \(|\psi\rangle\) is fun. of time generally \(\underline{|\psi(t)\rangle}\).

As system evolves \(|\psi(t)\rangle\) changes with time and rule of evolution describes the change

(3) \(\rightarrow\) Physical measurables are associated with "self adjoint" operators. i.e. \((\hat L, \hat p, x, \vec v, \cdots\) every observable\()\).

For every physical observable we have corresponding operator. not necessary independent with every; for ex. FOR k.E only velocity / momentum operator are will work.

To associate values with operator, we refer to eigen values of operators.

for operator

\[ A|\psi\rangle = \lambda|\psi\rangle \]

then '\(\lambda\)' is eigen value of \(A\) with eigen vector \(|\psi\rangle\).

For a given eigen value, if we have two or more linearly independent eigen vectors, then eigen values are repeated eigen values / degenerate eigen value.

It is also possible that no. of eigen values are infinite if LVS is infinite diamensional.

For \(n\times n\) matrix it can have max of \(n\) -- eigen values.

(4) Eigen values are in some sence "the only measurable values" for an operator cooresponding to physical observable should be real.

\(\rightarrow\) If \(A = A^{\dagger}\) (Hermitian conjugate) \(\left(\text{All eigen values are real for operator } A.\right)\)

i.e. if operator is self adjoint it's guaranteed that its all eigen-values are real.

Note :-- self adjoint are not equally same as Hermitian conjugate except in case of matrices.

Note : we can have contineous set of eigen values in case of infinite diamensional LVS.

For \(\underline{x}\), position of a particle can take any value for a definite momentum.

(4) The result of the measurement for any physical measurable observable is always an real eigen value for the corresponding operator.

\(\left\{\text{keep in mind every eigen value will be real for physical observable operators.}\right\}\)

Note :-- We can not say which eigen value will appear on measurment because of quantum uncertainty laws.

On the other hand we can repeatedly make measurments, take the average we call that mean value of the specific observable. or expectation value i.e. \(\underline{\langle x\rangle}\). and we have rule in Q.M to calculate expectation value. It also gives rule to calculate scatter about mean.

This can be achieved by making infinite identical copies of system and make measurements on each of them and take mean. of \(\langle x\rangle\).

Average may or may not be an eigen state of that operator.

\(\ast\) spectrum of an operator is the set of all eigen values of the operator. These eigen values can be contineous / discrete eigen values.

ex. the energy of \(e^-\) in H like atom goes like

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  \node[anchor=north] at (2.6,-1.0) {discrete};
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For Any operator \(A\), which acts upon \(|\psi\rangle\) state of system

let \(\lambda_1, \lambda_2 \cdots\) be the eigen values of \(A\) with eigenvectors \(|\phi_1\rangle, |\phi_2\rangle, \cdots\)

assuming \(|\phi_i\rangle\) are orthonormal basis \(\langle\phi_i|\phi_j\rangle = \delta_{ij}\)

\[ |\psi(t)\rangle = \sum_{n=1}^{\infty} c_n|\phi_n\rangle = \sum_{n=1}^{\infty} c_n(t)\,|\phi_n\rangle \]

\(\hookrightarrow\) assuming operators are time independent (explicitly).

\[ c_n(t) = \langle\phi_n|\psi\rangle \]

\(\hookrightarrow\) probability amplitude saying the actual state is \(|\phi_n\rangle\).

so \(c_n\) tells us how much of \(|\psi\rangle\) points along \(|\phi_n\rangle\).

\(\left\{|c_n(t)|^2 \text{ is the probability that the system is in the state } |\phi_n\rangle \text{ at time } t.\right.\)

if \(\langle\psi|\psi\rangle = 1\)

\[ \Rightarrow \quad \sum_n |c_n|^2 = \underline{1} \] \[ \langle A\rangle = \frac{\sum_n \lambda_n |c_n|^2}{\langle\psi|\psi\rangle} \Bigg/ \sum_n |c_n|^2 \] \[ A|\phi_n\rangle = \lambda_n|\phi_n\rangle \] \[ \langle\psi(t)|\psi(t)\rangle = \sum_n |c_n|^2 = \underline{1} \] \[ = \frac{\sum_n \lambda_n \langle\psi|\phi_n\rangle\langle\phi_n|\psi\rangle}{\langle\psi|\psi\rangle} \] \[ = \frac{\left\langle\psi\left|\sum_{n=1}^{\infty}\lambda_n|\phi_n\rangle\langle\phi_n\right|\psi\right\rangle}{\langle\psi|\psi\rangle} \] \[ = \left.\left\langle\psi\left|\sum_{n=1}^{\infty} A|\phi_n\rangle\langle\phi_n\right|\psi\right\rangle \right/ \|\psi\|^2 \] \[ = \frac{\left\langle\psi\left|A\sum_{n=1}^{\infty}|\phi_n\rangle\langle\phi_n\right|\psi\right\rangle}{\langle\psi|\psi\rangle} \] \[ \sum_{n=1}^{\infty}|\phi_n\rangle\langle\phi_n| = 1 \] \[ \boxed{\langle A\rangle = \frac{\langle\psi|A|\psi\rangle}{\langle\psi|\psi\rangle}} \qquad \text{or} \qquad \boxed{\langle A\rangle = \frac{\langle\Psi(t)|A|\Psi(t)\rangle}{\langle\Psi|\Psi\rangle}} \]

[\(\langle A\rangle\) is diagonal matrix element of \(\hat A\) in basis formed from \(\{|\Psi(t)\rangle\}\).]

for a normalized \(|\psi\rangle\), \(\langle\psi|\psi\rangle = 1\)

So,

\[ \boxed{\langle A\rangle(t) = \langle\psi|A|\psi\rangle} \]

where \(|\psi\rangle\) is time dependent \(|\Psi(t)\rangle\).

classical Physics

• Dyn. variables in phase space : \(q_1, q_2 \cdots\) ; \(p_1, p_2 \cdots\)
• PB relations \[ \{q_i, p_j\} = \delta_{ij} \] \[ 0 = \{q_i, q_j\} = \{p_i, p_j\} \]

\(H(q,p)\)

\[ \left. \begin{aligned} \dot q_i &= \frac{\partial H}{\partial p_i}\\[2pt] \dot p_i &= \frac{-\partial H}{\partial q_i} \end{aligned} \right\} \ \text{dynamics of variables} \]

such that,

\[ \begin{aligned} \{q_i(t), p_j(t)\} &= \delta_{ij}\\ \{q_i(t), q_j(t)\} &= 0\\ \{p_i(t), p_j(t)\} &= 0 \end{aligned} \]

Quantum Physics

• Dynamical variables are replaced by self adjoint operators (acting upon the vectors in the hilbert space).
• Phase space \(\xrightarrow{\ \text{replaced}\ }\) Hilbert space.
• PB's are replaced by commutator operators. \[ [A,B] = AB - BA \] \[ \frac{[q_i, p_j]}{i\hbar} = \frac{q_i p_j - p_j q_i}{i\hbar} = \delta_{ij} \] \[ [q_i, q_j] = 0 = [p_i, p_j] \]

Lec 6 \(>\) [Why we need \(i\hbar\) in \(\frac{[q_i,p_j]}{i\hbar} = \delta_{ij}\) relation ?]

\(\lozenge\) Commutator becomes anti-hermitian.

\[ A = A^{\dagger} \] \[ B = B^{\dagger} \] \[ [A,B]^{\dagger} = (AB - BA)^{\dagger} = B^{\dagger}A^{\dagger} - A^{\dagger}B^{\dagger} = -[A,B] = [B,A] \]

So

\[ [A,B]^{\dagger} = -[A,B] \]

since \(AB - BA\) is also a physical measurable, so we have to make sure its eigen values are real, which suggests we have to make it hermitian.

\[ \frac{[A,B]}{i\hbar} \ \text{ has real eigen values.} \]

'\(\hbar\)' is for diamentional reasons. such that \(\frac{[q_i, p_j]}{i\hbar}\) becomes diamensionless.

Now we know that

\[ \langle A\rangle(t) = \frac{\langle\psi(t)|A|\psi(t)\rangle}{\langle\psi(t)|\psi(t)\rangle} \]

The,

\[ \boxed{i\hbar\,\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle} \]

\(\left\{\text{operator eq}^{\text{n}}\text{ for vectors in Hilbert space}\right.\)

where \(H\) is an operator.

(postulate :-) statement :-

The time rate of change of state vector \(|\psi(t)\rangle\) is given by action of Hamiltonian operator on the state vector.

Note :-

\(|\psi(t)\rangle\) can be uniquely determined if we have initial state vector \(|\psi(0)\rangle\).

Since '\(H\)' is time independent, it appears that Shrodinger equation is autonomous in nature.

\(|\psi(t)\rangle\) can be realized by a coloumn vector where as \(H\) can be realized by a squarre matrix in Hilbert space.

\[ i\hbar\,\frac{d}{dt}|\psi(t)\rangle = H|\psi(t)\rangle \] \[ \frac{d}{dt}|\psi(t)\rangle = -\frac{iH}{\hbar}|\psi(t)\rangle \] \[ \boxed{|\psi(t)\rangle = \psi(0)\,e^{-\frac{iHt}{\hbar}}|\psi(0)\rangle} \] \[ \boxed{|\psi(t)\rangle = e^{-\frac{iHt}{\hbar}}|\psi(0)\rangle} \]

\(e^{-\frac{iHt}{\hbar}}\) is operator acts on \(|\psi(0)\rangle\)

Suppose \((t_0)\) is starting / initial time.

solution becomes

\[ \int_{t_0}^{t} \frac{d\big(|\psi(t)\rangle\big)}{|\psi(t)\rangle} = \int_{t_0}^{t} \frac{-iH}{\hbar}\,dt \] \[ \boxed{|\psi(t)\rangle = e^{-\frac{iH(t-t_0)}{\hbar}}\,|\psi(t_0)\rangle} \] \[ \langle\psi(t)| = \langle\psi(t_0)|\,e^{\frac{iH^{\dagger}(t-t_0)}{\hbar}} \]

(\(\rightarrow\) operator)

as eigen values of \(H\) to be real, \(H\) is Hermitian i.e. \(\underline{H = H^{\dagger}}\)

\[ \langle\psi(t)| = \langle\psi(t_0)|\,e^{\frac{iH(t-t_0)}{\hbar}} \]

so,

\[ \langle\psi(t)|\psi(t)\rangle = \langle\psi(t_0)|\,e^{\frac{iH(t-t_0)}{\hbar}}\cdot e^{-\frac{iH(t-t_0)}{\hbar}}\,|\psi(t_0)\rangle \] \[ = \ ? \] \[ \left\{ \begin{aligned} e^{A}\cdot e^{B} &\neq e^{A+B} \quad \text{for } [A,B] \neq 0\\ e^{A}\cdot e^{-A} &= e^{0} = \mathbb{1} \ \text{operator} \end{aligned} \right. \]

\(\hookrightarrow\) true always. as \(\{A, -A\} = 0\)

\[ = \langle\psi(t_0)|\mathbb{1}|\psi(t_0)\rangle \] \[ \boxed{\langle\psi(t)|\psi(t)\rangle = \langle\psi(t_0)|\psi(t_0)\rangle} \]

conservation of norm of state vector.

The analogue of conservation of phase space volume element is reflected in conservation of norm of state vector \(|\psi(t)\rangle\). It is because of fact that the Hermitian conjugate of operator \(e^{\frac{iH(t-t_0)}{\hbar}}\) is its inverse.

lets call

\[ e^{-\frac{iH(t-t_0)}{\hbar}} = U(t, t_0) = \text{evolution operator} \]

or the time development operator. (unitary). or propogator.

Such operator \(U(t, t_0)\) has interesting properties.

\[ U^{\dagger} = U^{-1}(t, t_0) \]

(or)

\[ UU^{\dagger} = U^{\dagger}U = I \qquad \text{--- unitary matrix} \]

If system has \(t_1\) time as initial time between \(t_0\) and \(t_2\)

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\[ U(t_2, t_0) = U(t_1, t_0)\,U(t_2, t_1) \]

semi -- group property.

Q) If find \(f(t)\).

\[ \frac{d}{dt} f(t) = H(t) f(t) \] \[ \Rightarrow \quad \frac{d f(t)}{f(t)} = H(t)\,dt \] \[ \Rightarrow \quad \boxed{f(t) = e^{\int_0^{t} H(t')\,dt'}\ f(0)} \]

\(\rightarrow\) Not the formal sol\(^{\text{n}}\) as

\[ e^{H(t_1) + H(t_2)} \neq e^{H(t_1)}\cdot e^{H(t_2)} \] \[ \big(H(t_1), H(t_2)\big) \neq 0 \]

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