Quan lect 4

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Section I

Quantum Mechanics

Lecture 1.4

Books:

(

1)

A. Beiser, “Concepts of Modem Physics”,

McGraw Hill

(2) K S Krane, “Modern Physics”, John Wiely & Sons, Inc.,

3

rd

_{Ed. (2011).}

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Wave function, ψ: The quantity whose variation makes matter wave is know as wave function.

Probability of finding particle lies between 0 (object is not there) to 1 (object is definitely there).

Suppose probability is 0.2 i.e. 20 %.

…means you have 20 % chances of finding particle but amplitude of wave can be negative .

So, wave function, ψ, has no physical significance.

However, the square of the wave function, |ψ|2_{is always}

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The wave functions are usually complex with both real and complex Parts. The probability is however is always a positive quantity.

The probability density |Ψ|2_{is defined as the product Ψ Ψ}*_{of Ψ and its}

complex conjugate Ψ*.

Wave function:

Its complex conjugate is:

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Normalisation of wave function: A wave function is said to be Normalized if it satisfy the following condition

Since the square of wave function give the probability of finding the Particle it means according to above equation particle is definitely there.

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Well behaved or acceptable wave function:

(1) Since from the wave function we find the probability of finding the particle at particular point of space and time which should

have unique value and therefore a well behaved wave function must be single valued and continuous

(2) must also have continuous and single value.

(3) The wave function must be normalized . It means

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Schrodinger Wave Equation:

Here we shall write a wave equation in terms of wave function Ψ. The solution of that wave equation helps us in known the value

of wave function and hence different physical quantities we can find from wave function as function of space or time.

We shall discuss:

Schrodinger Time Dependent Wave Equation.

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Schrodinger Time Dependent Wave Equation:

To get the Schrodinger’s time dependent wave equation we consider the wave function Ψ which describe the particle of following form

---(1)

Note that we considered the complex wave function. Above

equations says that the motion is in x-direction and v is the velocity of the particle.

Using ω = 2πν and υ = νλ, above Eq. becomes,

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Now using the following relations

---(3) Eq. (2) becomes

---(4)

where E is the energy of the particle and p is the momentum of the particle.

Eq. (4) describe a free particle which is not restricted

by any external force. But in actual problem we deals with the particle Motion which are restricted by some kind of force. For example

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To deal with such cases we need to find a differential Eq. for the wave function Ψ whose solution will give us Ψ according to

Desired situation.

To get the form of this Eq. we differentiate Eq. (4) w.r.t x twice . So we have

---(5)

Differentiate Eq.(4) w.r.t. t , we get

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At speed small compared to the speed of the light, the total energy of the particle is the sum of K.E. and potential energy

We have

---(7) Note the P.E. is function of x and t.

.

Multiplying Eq. (7) by Ψ, we get

_ ---(8) Using (5) and (6) in (8) we get,

---(9)

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In three dimension above Eq. is written as

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Expectation values: Once we solved the Schrodinger Wave Eq. we Get the value of the wave function. The next step will be to

get the knowledge about the physical quantities like position, Momentum, energy etc. from the wave function.

We find the expectation values of the desired physical quantities From the wave function.

For example the expectation value of position is

---(1)

If the wave function is normalised then denominator of above equation will be unity, so we have

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For a general function G(x) e.g. U(x) we write the expectation value as

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Operators (Another way to find the expectation values):

Here we shall discuss that how we find the expectation value of and .

We write the following Eq. for the wave function

---(1) Where E is the energy and p is the momentum of the particle. Now we differentiate Eq. (1) w.r.t. x and t

---(2)

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Now we write above Eq. as

---(4)

---(5)

Above Eqns. Tells is that the dynamical quantities p and E are related to the differential operators and

Respectively.

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So, we write the momentum operator and energy operator as

---(6)

---(7)

Total energy of a particle is written as

---(8) Where KE is

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Using (6) we write Eq. (9)

---(10)

Using (7) and (10), Eq (8) become

---(11)

Operating above operator on wave function we get

---(12)

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The expectation value of the momentum and energy is written as

---(13)

---(14)

In general for a quantity G(x,p) we write the expectation value as

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Schrodinger Time independent wave Equation: In many situations the potential energy of the particle does not depend upon the time but

depends only on the position of the particle. In such cases we write the Time independent Schrodinger wave Eq.

To get this Eq. first we write the wave function for a unrestricted particle as

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We know the time dependent wave Eq. is

---(2)

Using Eq. (1) in (2) we get,

---(3)

Note that in above Eq. the exponential factor is common, so we divide with this factor and we write above Eq. as

---(4)

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(23)

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Eigen values and Eigen function: There are certain dynamics quantities which can take only certain values. Suppose G is such physical quantity. Now when the operator of G is operated on the wave function

, we get the discrete values .

So we write

---(1)

Above Eq. is know as Eigen value Equation. The wave function is known as Eigen wave function and is known as corresponding Eigen values.

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Exercise: