Eigenfunction proof and eigenvalue

AI Thread Summary
The discussion revolves around the confusion regarding the proof of an expression as an eigenfunction. The user initially struggles to find the necessary formulas and seeks clarification on what constitutes an eigenfunction. After some contemplation, they realize that the solution is straightforward, requiring only multiplication and differentiation. This highlights the importance of understanding basic concepts in eigenfunctions and their properties. Ultimately, the user resolves their query with a simple approach.
Andrei0408
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Homework Statement
Prove that x*e^(-x^2/2) is eigenfunction for the operator x^2 - (d^2/dx^2) and find the corresponding eigenvalue.
Relevant Equations
I don't know what equations I need to use
I searched through the courses but I can't find any formula to help me prove that the expression is an eigenfunction. Am I missing something? What are the formulas needed for this problem statement?
 
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You want to know: what is an eigenfunction?

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PeroK said:
You want to know: what is an eigenfunction?

Internet search?
Yeah nevermind, it was very trivial, just had to multiply and derivate, for some reason I thought I needed something else
 
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Thread 'Variable mass system : water sprayed into a moving container'

Thread 'Variable mass system : water sprayed into a moving container'

Starting with the mass considerations #m(t)# is mass of water #M_{c}# mass of container and #M(t)# mass of total system $$M(t) = M_{C} + m(t)$$ $$\Rightarrow \frac{dM(t)}{dt} = \frac{dm(t)}{dt}$$ $$P_i = Mv + u \, dm$$ $$P_f = (M + dm)(v + dv)$$ $$\Delta P = M \, dv + (v - u) \, dm$$ $$F = \frac{dP}{dt} = M \frac{dv}{dt} + (v - u) \frac{dm}{dt}$$ $$F = u \frac{dm}{dt} = \rho A u^2$$ from conservation of momentum , the cannon recoils with the same force which it applies. $$\quad \frac{dm}{dt}...
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