What Is the E/U Ratio for a Half Reflection Coefficient in Quantum Tunneling?

AI Thread Summary
To determine the E/U ratio for a reflection coefficient of 1/2 in quantum tunneling, the relationship between transmission and reflection coefficients is crucial. The reflection coefficient is defined as (k1-k2)^2 / (k1+k2)^2, where k1 and k2 depend on the particle's energy and the barrier height. The discussion highlights the challenge of isolating a numerical ratio without specific values for mass and energy, as these variables are intrinsic to the equations. Participants suggest expanding the reflection coefficient to explore potential simplifications. Ultimately, the problem emphasizes the complexity of deriving a simple numerical ratio from the given quantum mechanics equations.
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Homework Statement


A particle of energy E approaches a step barrier of height U. What should be the ratio E/U be so that the reflection coefficient is 1/2.


Homework Equations


Transmission + Reflection = 1
Transmission = e^(-2a*alpha)
a = mw∏/h
alpha^2 = (8m∏^2/h^2)(U-E)

Reflection = (k1-k2)^2 / (k1+k2)^2



The Attempt at a Solution


So this is a little bit messy.. I assume most people that are going to help me are familiar with these basic transmission and reflection equations. Anyway is there a possible way to solve this problem an get a ratio that is JUST a number? wouldn't you always have other variables in it since it's talking about a completely undefined particle? there are no ways to get the m's and w's out of the equations right?
 
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All those k's depend on energy.
Expand them out in the reflection coefficient and see what happens.
 
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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