Why Are There Two Different Expressions for Uncertainty in Quantum Mechanics?

AI Thread Summary
The discussion centers on the two expressions for the uncertainty principle in quantum mechanics: Del(X) x Del(P) ≥ h/2π and Del(X) x Del(P) ≥ h/4π. The first expression is traditionally associated with Heisenberg's principle, while the second is derived from Fourier analysis and is favored in modern textbooks. The lecturer indicated that both can be used, but the second expression is more relevant for competitive exams. In one-dimensional scenarios, the lower bound of the uncertainty product aligns with the second expression. Understanding these distinctions is crucial for exam preparation in quantum mechanics.
Amith2006
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Homework Statement


According to Heisenberg Uncertainty principle,
Del(X)xdel(P) >= h/2(pi)
But there is also another expression given in my book which is,
Del(X)xdel(P) >= h/4(pi)
I asked my lecturer why is it so? She said that, the second expression is obtained from Fourier analysis. But she also said that in calculations, you can use the first one. Which should I use in competitive exams?



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The Attempt at a Solution

 
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The second one is the form given in all modern textbooks and the one you will be expected to use in your exams.
 
In 1D I believe the lower bound on the product of these uncertainties is your second expression.
 
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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