1. Linear Vector Space and Matrix Mechanics
Chapter 1
Lecture 1.14
Dr. Arvind Kumar
Physics Department
NIT Jalandhar
e.mail:

2. 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)

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

4. 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)

5. 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)

6. 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)

7. 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.

8. 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.

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

10. 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

11. 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)

12. 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.

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

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

15. 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.

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

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

18. 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)

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