Normalization of a wavefunction

AI Thread Summary
The discussion centers on the normalization of wavefunctions in quantum mechanics, specifically addressing the factor of "√2/a" in the normalized eigenfunctions. It clarifies that the wavefunction should be expressed in terms of normalized eigenfunctions, denoted as φ_n(x), rather than unnormalized sine functions. The normalization factor arises to ensure that the total probability is equal to one, which is essential for valid wavefunctions. Participants confirm the importance of using normalized forms for clarity and accuracy in calculations. The conversation concludes with a question about the coefficients in a specific wavefunction, indicating ongoing confusion regarding normalization.
tina21
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Homework Statement
Normalize the wave function
Ѱ(x,0)=A/√a sin(ᴨx/a) +√3/5 sin(3ᴨx/a) + 1/√5a sin(5ᴨx/a)
Relevant Equations
Ѱ(x,0)=A/√a sin(ᴨx/a) +√3/5 sin(3ᴨx/a) + 1/√5a sin(5ᴨx/a)
I tried writing the function as:

Ѱ = c1Φ1 + C2𝚽2 + C3𝚽3

in order to then find mod C1^2...

But ɸ = √2/a sin(ᴨx/a) and not sin(ᴨx/a)

I cannot understand how the factor of "√2/a " comes
 
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tina21 said:
I cannot understand how the factor of "√2/a " comes
What do you not understand about $$\phi_n(x) = \sqrt{\frac 2 a}\sin(\frac{n\pi x}{a})$$
 
PeroK said:
What do you not understand about $$\phi_n(x) = \sqrt{\frac 2 a}\sin(\frac{n\pi x}{a})$$
According to me the function shouldn't have had the factor √2/a but I now believe the factor arises upon normalising phi (x). Is that correct?
 
tina21 said:
According to me the function shouldn't have had the factor √2/a but I now believe the factor arises upon normalising phi (x). Is that correct?
Yes. ##\phi_n(x)## is, by definition, a normalised wavefunction. For that reason, it is always best to organise things so that you have $$\psi(x) = \sum a_n \phi_n(x)$$ and not $$\psi(x) = \sum a_n \sin (\frac{n \pi x}{a})$$ From that point of view, the question has made things a little difficult for you - but it should be easy enough to take the first step and express your wavefunction in terms of normalised eigenfunctions.
 
PeroK said:
Yes. ##\phi_n(x)## is, by definition, a normalised wavefunction. For that reason, it is always best to organise things so that you have $$\psi(x) = \sum a_n \phi_n(x)$$ and not $$\psi(x) = \sum a_n \sin (\frac{n \pi x}{a})$$ From that point of view, the question has made things a little difficult for you - but it should be easy enough to take the first step and express your wavefunction in terms of normalised eigenfunctions.
Thank you. I now understand.
 
tina21 said:
Homework Statement:: Normalize the wave function
Ѱ(x,0)=A/√a sin(ᴨx/a) +√3/5 sin(3ᴨx/a) + 1/√5a sin(5ᴨx/a)
Relevant Equations:: Ѱ(x,0)=A/√a sin(ᴨx/a) +√3/5 sin(3ᴨx/a) + 1/√5a sin(5ᴨx/a)
Shouldn't the coefficient in front of the second term be ##\sqrt{\dfrac{3}{5a}}##?
Just asking ##\dots##
 
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