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Q35E

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Modern Physics
Found in: Page 281
Modern Physics

Modern Physics

Book edition 2nd Edition
Author(s) Randy Harris
Pages 633 pages
ISBN 9780805303087

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Short Answer

A mathematical solution of the azimuthal equation (7-22) is Φ(φ)=Ae-+Be- , which applies when D is negative, (a) Show that this simply cannot meet itself smoothly when it finishes a round trip about the z-axis. The simplest approach is to consider φ=0 and φ=2π. (b) If D were 0, equation (7-22) would say simply that the second derivative Φ(φ)of is 0 . Argue than this too leads to physically unacceptable solution, except in the special case of Φ(φ) being constant, which is covered by the ml=0 , case of solutions (7-24).

(a) The given function cannot meet itself smoothly when it finishes a round trip about the z-axis.

(b) If D = 0, the equation Φ(φ)=Ae-+Be- will be a constant and will be a linear function, it will only repeat itself after 2π if the slope is zero.

See the step by step solution

Step by Step Solution

Step 1: A concept:

Azimuthal quantum number specifies shape and angular momentum of the orbital.

Step 2: (a) Values of the equation at  φ=0 and φ=2π :

Consider the given data as below.

Φ(φ)=Ae-+Be- ….. (1)

Where, is the Azimuthal Angle, are the Arbitrary constants, and is the Azimuthal function.

D=-1Φ2Φφ2

Let, eq. (1) is continuous when φ=0 and φ=2π.

As you know that, for a function to be continuous at φ=0 and φ=2π , the value at φ=0 and shφ=2π ould be equal.

Hence, for that to hold,

Φ0=Φ2πAe-D×0+Be--D×0=Ae-D×2π+Be--D×2πA+B=Ae-D2π+Be--D2π ….. (2)

Also, if its derivative is continuous

-DA-B=-DAe-D2π+Be--D2πA-B=Ae-D2π+Be--D2π ….. (3)

Now, by adding equation (2) and (3), you get,

A=Ae-D2π ….. (4)

Also, by subtracting equation (3) from (2), you get,

B=Be--D2π ….. (5)

Step 3: Conclusion:

For the equation (4) to hold,

Either A=0 or D=0

And for eq. (5) to hold

Either B=0 or D=0

You can’t have both A=0 and B=0 , wave function will not be possible if it holds.

And if D=0 , the given equation Φφ=Ae-Dφ+Be--Dφ will be a constant.

Hence, the given function cannot meet itself smoothly when it finishes a round trip about the z-axis.

Step 4: (b) simply that the second derivative of Φ(φ) is 0 :

If D=0 , the equation Φφ=Ae-Dφ+Be--Dφ will be a constant and will be a linear function, it will only repeat itself after 2π if the slope is zero.

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