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3.1 Particles and barriers

Slides: Video 3.1.3 Orthogonality offunctions

Text reference: Quantum Mechanicsfor Scientists and Engineers

Section 2.7 (“Orthogonality ofeigenfunctions” and “Expansion

coefficients”)

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Particles and barriers

Orthogonality of function

Quantum mechanics for scientists and engineers Da

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Orthogonal functions

In addition to being a complete set

the eigenfunctions

are also “orthogonal” to one anotherTwo non-zero functions and

defined from 0 to  L z

are said to be orthogonal if

  2sin

n

 z z

n z z

 L L

    

g z   h z

0

0

 z L

g z h z dz

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Orthogonal functions

It is easy to show mathematically that thefunctions

are orthogonal to one anotherbecause, as we could prove

for

This mutual orthogonality is another commproperty of the eigenfunctions we will find

  2 sinn

 z z

n z z L L

    

0

0

 z L

n m z z dz     n m

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Orthogonality and parity

Functions with opposite parity

such as the first two particle-in-abox eigenfunctions

are always orthogonal

Whatever contribution we get to theintegral from the left side

we get the opposite from the rightso they cancel out

we do not even need to workout the integral

1n 

2n 

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Orthogonality and parity

With sine functions like this

even those sine functions of thesame parity are also orthogonal

which is less obvious but also

trueFor example

the n=3 eigenfunctionis orthogonal to

the n=1 eigenfunction

1n 

2n 

3n 

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Orthonorcondit

Orthonormality

We can introduce the “Kroneckerdelta”

For our normalized eigenfunctions

we can therefore write theorthogonality and thenormalization as one condition

A set of functions that is bothnormalized and mutually

orthogonal is “orthonormal”

0

 z L

n m z

0

1

fonm

nn

 

 

Kronecke

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Expansion coefficients

Suppose we want to write the function

in terms of a complete set of orthonorma

functionsi.e., we want to write

To find the “expansion coefficients” cnpremultiply by

and integrate

 f x

n   x 

n n

n

 f x c x 

m  x 

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Expansion coefficients

Premultiplying by and integrating giv

If we have an orthonormal complete setwe can now expand any function in it

Generally an integral like

is also called an “overlap integral”

m  x 

m m n nn x f x dx x c x dx  

 

n m n

n

c x x dx    

 

mc

m

  x f x dx 

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