Linear Vector Space and Matrix Mechanics Chapter 1 Lecture 1.14 Dr. Arvind Kumar Physics Department NIT Jalandhar e.mail: iitd.arvind@gmail.com https://sites.google.com/site/karvindk2013/

Time evolution of the system state: Given initial state , how to find the state at later time t. Let consider a linear operator , We write -------(1) Time evolution operator or propagator. Note that at initial time t0, we have identity operator -------------------(2)

Now we find time evolution operator. Consider Schrodinger Eq. --------(3) Using (1) in (3), we get -----(4) ------------(5)

Now if the Hamiltonian is independent of time, then integration of (5) and also using the initial condition (2), we get ----------(6) is unitary operator ----------(7)

Stationary states: time independent potential Schrodinger time dependent Eq. in position representation is -----(1) If we have time independent potential ---------(2) Using separable technique, we write --------(3)

Using (3) in (1) and dividing by ----------(4) L.H.S is function of time only whereas R.H.S function Of position, so equate both to a constant E, which Has dimensions of energy. -----------(5)

And -------------(6) Which is time independent Schrodinger Eq. Solution of (5) is ---(7) Using (7) in (3), we get ----------(8) Which is solution of (1), for time independent Potential.

Eq. (8) is called stationary states. The probability density is stationary. It does not depend upon time. --------(9) Energy level which are sol. to eq. (6) are called energy spectrum of system. States corresponding to discrete and continuous spectra are called bound and unbound states.

Most general solution to (1) can be written as Where,

Exercise: Show that the probability density does not evolve in time: = 0 2. Derive the Eq. for conservation of probability Density Probability density Probability current density

Conservation of probability density: Time dependent Schrodinger Eq, --(1) Taking complex conjugate, ------(2) Multiply both sides of (1) by and both sides Of (2) by and subtract, we get --------(3)

Writing (3) as, ----(4) Where, -------(5) is Probability density . is Probability current density Or current density or particle density flux. Eq. (5) is known as conservation of probability.

Time evolution of expectation value: For normalized state, we write the expectation of operator as -------(1) Using ------(2) We can write ---------------------(3)

Using (3), we can write from (1), time evolution of expectation vale of operator as

The energy, linear momentum and angular momentum of an isolated system are conserved In classical physics, conservation of energy, linear momentum and angular momentum are consequences of homogeneity of time, homogeneity of space and Isotropy of space respectively. Here in quantum mechanics above symmetries are associated with invariance in time translation, space translation, and space rotation.

Schordinger Picture: Here state vectors depend explicitly on time but Operators do not. We already discussed time evolution of state vector and given by

Heisenberg Picture: Here time dependence of state vector is frozen. ---------(1) Using ----(2) We write -----(3) is frozen in time -------(4)

Evolution of the expectation value of an operator --------(5) Where -----(6) Eq. (5) show that the expectation values of operator Is same in both pictures. At t = 0 both pictures concides. -----(7) And -------------(8)

Exercise: Using (6), derive Heisenberg equation of Motion ----(9)