Linear Vector Space and Matrix Mechanics Chapter 1 Lecture 1.15 Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: iitd.arvind@gmail.com https://sites.google.com/site/karvindk2013/

Harmonic oscillator: Classic harmonic oscillator is a mass m attached to Spring of force constant k. Hooke’s Law, ----(1) Solution: ------(2) Where, Angular frequency, -----(3)

Potential energy: Parabola -----(4) The sum of the kinetic and  potential energies in a simple harmonic oscillator  is a constant, i.e., KE+PE=constant. The energy oscillates back and forth between kinetic and potential, going completely from one to the other as the system oscillates.

Any potential is approximately parabolic in the neighbourhood of a local minimum. Expanding V(x) in a Taylor series about minimum --------(5)

Subtracting from potential given in Eq. (5) will not effect as adding constant does not effect force. As x0 is minimum, so . Also neglecting higher order terms as long as is small. So we get from (5), -----(6) Above Eq describe simple harmonic oscillation about x0 with spring constant Any oscillatory motion is approximately simple Harmonic as long as amplitude is small.

Quantum Harmonic oscillator: Hamiltonian operator for a particle of mass m which oscillate with an angular frequency ω under the one dimensional harmonic potential is -------(1) Two methods to solve: Power series method in which one solve Schrodinger Eq Ladder or Algebraic method where we use operator algebra (Matrix method)

Algebraic method: Consider two dimensionless Hermitian operators ------(2) We write Hamiltonian of Eq (1) now using above operators, ------(3) Introducing two dimensionless, non-hermitian Operators, ------(4)

Note that, -------(5) Where, ----(6) Hence --------(7) -------(8)

Using (8) in (2), --------(9) Number operator or occupation number operator. It is Hermitian. Also, i ------------(10) So ----------- (11)

Energy Eigenvalues: Hamiltonian is linear in number operator and thus both commute and thus have simultaneous Eigenstates. We write -----------(12) and -----------(13) is energy eigenstates. Using (9) and (13), we get ------------(14) n is positive integer. We will prove it!

Physical meaning of : Note that , Using (9) and (11) ----------(15)

Using (13) and (15), we cna write --------(16) -------(17) Note from above, and are eigenstates of with eigenvalues respectively. Action of is to generate new energy states that lower and higher energy by one unit .

= Lowering operator or anihilation operator = Raising operator or creation operator Also know as ladder operators. Now we prove the following:

Using the identity and , we can write ------(18) Hence, Combining with ,we get --(19) ---(20)

Thus we can write ---------(21) Cn is a constant. is normalize for all values of n ---------(22) Also --------(23)

From (22) and (23), we get ------(24) Note that n cannot be negative. . Using (24) in (21), we get ---------(25)

Show that ----(26) The energy spectrum of harmonc oscillator is Discrete. --------(27)

Energy Eigenstates: Using (26), we can write the eigenvectors as, -------(28)

Orthogonality and completeness condition: form complete and orthonormal basis.3 --------(29)

Energy eigenstates in position space: In position representation, we write the operator as, -------(30) Where, . Dimension of length. Annihilation and creation operators are written as -------(31) -------(32)

Using (31), in position space is written as, -------(33) ------(34) Where . Solution of (34) is, ------------(35)

Normalizing we get the constant A ----(36) -----(37) Ground state wave function is --------(38) Which is a Gaussian function.

For 1st excited state

For 2nd and 3rd excited state:

For nth state, we write in general, ------------(39)

The probability of finding the particle outside the classical allowed range is non-zero. Also in odd states the probability of Finding the particle is zero at the centre.

When n is very large (say 100) then the quantum picture resemble the classical case.

Matrix representation of various operators:

Expectation values of operators and Heisenberg Principle:

From above two equations we conclude that the expectation value of potential energy is equal to the expectation value of K.E. And both are equal to the half of total energy.

Heisenberg Uncertainty relation: