Linear Vector Space and Matrix Mechanics Chapter 1 Lecture 1.13 Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: iitd.arvind@gmail.com https://sites.google.com/site/karvindk2013/

Complete set of commutating observables: Compatible observables: Two observables are compatible if corresponding operators commute If operator do not commute then we call them non-compatible. If two operators do not commute then the corresponding observables cannot be measured simultaneously. The order in which they are measured matters. However if they commute then order of measurement does not matter.

If two observables are compatible then their corresponding operators possesses a set of Common or simultaneous eigenstates (both for degenerate and non-degenerate case). Proof: Let is non-degenerate state is joint eigenstate of operators and .

Let simulatenous eigenstates are denoted by Theorem can be generalized to many set of mutually compatible observables. And these posses joint eigenstates Completeness and orthonormality condition are,

If operator has degenerate eigenvalues then the specification of eigenvalue does not uniquely determine the state of system. For example: Free particle in one dimension, eigenvalues of hamiltonian are double degenerate Eigen values Eigen functions: and For free particle the momentum eigen value is and corresponding eigen function is . Merely specifying the energy of the particle does not uniquely specify the state of system. However specifying the momentum help in determining the state of system completely.

Among the degenerate eigenstates of only a subset will be eigenstates of B. Thus the set of states that are joint Eigenstates of both operators is not complete. To resolve degeneracy we introduce a third operator C which commute with A and B. We can then construct a set of Joint eigenstates of A, B and C which is complete. If degeneracy still eixit we introduce a new operator.

Continuing in this way, we will finally obtain complete set of commutating operators (CSCO). A set of hermitian operators is called CSCO, if the operators mutually commute and if the set of their common eigenstates is complete and not degenerate (i.e. Unique)

Heisenberg’s Uncertainty principle: In classical mechanics the particles position and momentum can be determined precisely if we know the initial conditions. However in wave mechanics we describe the particles by the wave packet. The particle can be located anywhere with this wave packet. The position of the particle becomes more and more definite as the size Of the wave packet becomes smaller and smaller (see fig) However in such situation the average value of the wavelength becomes uncertain because in small wave packet we have small number of wavelengths present. Since wavelength is related to the momentum and thus momentum also becomes uncertain.

However if we consider the opposite case, i.e. the size of the wave packet is larger then the wavelength can be determined precisely as we have more no of waves inside the wave packet and hence the momentum of the particle can also be determined precisely. However In this situation the position of the particle becomes uncertain. Thus we conclude that, It is impossible to measure precisely and simultaneously both the position and momentum of the particle to any desired degree of accuracy. If Δx is the uncertainty in measurement of the position and Δp is the uncertainty in measurement of momentum of the particle then ----------(1)

A wave with well defined wavelength but ill defined position A wave with well defined position but ill defined wavelength Uncertainties Δx and Δp are known as standard Deviations. Standard Deviation

Consider two hermitian operators Expectation values ------(1) Derivation of general uncertainty relation: Consider two hermitian operators Expectation values ------(1) Now consider operators -----(2) We have ----------(3)

We write the expectation values -------(4) Where, ----------(5) The uncertainties are defined by ------------(6)

We operate the operators and on state ---------(7) Using the Schwartz inequality we get . ------------(8) ---------(9)

Using (8) and (9), we get ----(10) We can write ---------(11) Where, we used . antihermitian Hermitian Expectation values of Hermitian operator are real and That the anti-hermitian imaginary

Expectation value is sum of real and imaginary part . Hence, -----------(12) Last quantity is real positive, so we can write -----(13) Using (10) and (13), we get -----(14)

Eq. (14) further written as ----(15) Which is uncertainty relation. For x- component of postion and momentum operator We can write,