Dummy indices in Wave packet equation

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Zinc Potterman - 16 May 2005 10:13 GMT

In my quantum mechanics text book it states that if the normalisation
condition for a wave
packet Psi made of a superposition of states is

Psi*Psi=1

then the two occurences of Psi necessitate two different dummy indices in
the sum.
m amd n for example.
Is this a general thing in maths that if a function contains a sum then a
product of that function (with its complex conjugate in this case)requires 2
separate indices.
Regards
Zinc

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Jon Bell - 16 May 2005 13:29 GMT

>Is this a general thing in maths that if a function contains a sum then a
>product of that function (with its complex conjugate in this case)requires 2
>separate indices.

Yes. To see this, simply write out the sum(s) and perform the indicated
multiplication. Use a a small, finite number of terms in the sum to make
the algebra manageable. For example, let f = f_1 + f_2 + f_3. Then

f*f = (f_1 + f_2 + f_3)*(f_1 + f_2 + f_3)

f*f = f_1*f_1 + f_1*f_2 + f_1*f_3
+ f_2*f_1 + f_2*f_2 + f_2*f_3
+ f_3*f_1 + f_3*f_2 + f_3*f_3

Writing the product using f in summation form, if you try to use a single
index you get

f*f = sum(f_n*f_n)

f*f = f_1*f_1 + f_2*f_2 + f_3*f_3

which is very different from what I wrote above. On the other hand if you
have two indices, that indicates a "double sum" (even though we may write
it as a single summation for brevity):

f*f = sum over m (sum over n (f_m*f_n))

f*f = sum over m (f_m*f_1 + f_m*f_2 + f_m*f_3)

f*f = f_1*f_1 + f_1*f_2 + f_1*f_3
+ f_2*f_1 + f_2*f_2 + f_2*f_3
+ f_3*f_1 + f_3*f_2 + f_3*f_3

which is what I wrote first above.

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Jon Bell <jtbell@presby.edu> Presbyterian College
Dept. of Physics and Computer Science Clinton, South Carolina USA

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Natural Science Forum / Physics / Particle Physics / May 2005

Dummy indices in Wave packet equation