PYL-100-2016-QMLect-02-ProbContEq

8/18/2019 PYL-100-2016-QMLect-02-ProbContEq

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PYL100 Electromagnetic Waves and Quantum Mechanics

Continuous Variables

• If I select a random person off the street, the probability that her age is

precisely 16 years, 4 hours, 27 minutes, and 3.333 ... seconds is zero.

• The only sensible thing to speak about is the probability that her age lies in

some interval-say, between 16 and 17.

• If the interval is sufficiently short, this probability is proportional to the

length of the interval.

probability density

The probability that x lies between a and b (a finite interval) is given by

the integral of (x):


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Continuous VariablesContinuousDiscrete


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Problem

Determine A


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NORMALIZATION

is the probability density for finding the particle at point x , at

time t

Is it consistent with Schrödinger equation ?

Schrödinger equation:

(x. t) is a solution A (x. t) is also a solution

You can choose A such that (x. t) satisfy the normalizing condition

Normalizing the

wave function


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NORMALIZATION

is the probability density for finding the particle at point x , at

time t

If the integral goes to Infinity or zero?

• Such non-normalizable solutions cannot represent particles, and

must be rejected.

Physically realizable states correspond to the square-integrable

solutions to Schrödinger equation


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TIME DEPENDENCE OF NORMALIZATION

is the probability density for finding the particle at point x, at

time t

Suppose I have normalized the wave function at time t = 0.

How do I know that it will stay normalized, as time goes on ?


7/15PYL100 Electromagnetic Waves and Quantum Mechanics


Schrödinger equation

Conjugate




(x . t) must go to zero as x goes to () infinity-otherwise the

wave function would not be normalizable.

(x . t) is normalized at t = 0, it stays normalized for all future

time

• Schrödinger equation has the remarkable property that it

automatically preserves the normalization of the wave function.

• Without this crucial feature the Schrödinger equation would be

incompatible with the statistical interpretation, and the whole

theory would crumble



Probability current

The probability of finding a particle in the range (a < x < b), at time t.

• J : the rate at which probability is "flowing" past the point x .

•

J has the dimensions 1/time, and units seconds−1

.

= (, )

, =

Define probability current


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Continuity Equation in Electrodynamics

Charge density Current density

Integrating over a small volume we have

• This shows that any decrease in the charge, in an infinitesimal

volume, must be accompanied by a flux of electric current out ofthe volume.

• In other words, charge has to be locally conserved.

• The total charge of a system is a constant independent of time


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Continuity Equation

The Schrödinger equation

in three-dimension

and its complex conjugate

Continuity equation in

Quantum mechanics

( )× Ψ∗

( )× Ψ

Subtracting we get,

Where,

Can be written asWhere, Position probability density

Probability current density


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Example

Find the probability current for the wave function:

Answer:


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Examples

Ψ , = (−)

, = →

= 0

Find probality current J (x, t)

=ℏ

2 =

ℏ

=

Plane wave function

Consistent with Electrodynamics


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Expectation Value

• is the average of measurements performed on particles

all in the state or the average of repeated measurements

on an ensemble of identically prepared (all in state )

systems

• It does not mean that if you measure the position of one

particle over and over again and the average is theexpectation value.

• It is not the most probable value.


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Expectation value of the velocity

Postulate: expectation value of the velocity is equal to the time derivative of the expectation

value of position

“Velocity" of ⟨⟩

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PYL-100-2016-QMLect-02-ProbContEq

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