Consider a particle bound in a infinite well, where the potential inside is not constant but a linearly varying function. Suppose the particle is in a fairly high energy state, so that its wave function stretches across the entire well; that is isn’t caught in the “low spot”. Decide how ,if at all, its wavelength should vary. Then sketch a plausible wave function.
The graph is plotted with energy and potential inside a potential well.
If potential rises higher than the particles total energy then the particle is stuck in the potential well and it oscillates between the turning points and it cannot escape. Hence the turning points in this condition is located thus the oscillator wave function stretches across the entire well with quantum states with node and antinode.
As the kinetic energy varies inversely with wavelength, amplitude becomes larger at the regions where kinetic energy is smaller and the wavelength will be shortened for region where kinetic energy is larger
For high potential inside a well , the particles wave function tunnels through the finite potential barrier and is finally brought to zero.
the plausible wavefunction is
In a study of heat transfer, we find that for a solid rod, there is a relationship between the second derivative of the temperature with respect to position along the rod and the first with respect to time. (A linear temperature change with position would imply as much heat flowing into a region as out. so the temperature there would not change with time).
(a) Separate variables this assume a solution that is a product of a function of x and a function of t plug it in then divide by it, obtain two ordinary differential equations.
(b) consider a fairly simple, if somewhat unrealistic case suppose the temperature is 0 at x=0 and , and x=1 positive in between, write down the simplest function of x that (1) fits these conditions and (2) obey the differential equation involving x. Does your choice determine the value, including sign of some constant ?
(c) Obtain the full for this case.
Show that that is, verify that unless the wave function is an Eigen function of the momentum operator, there will be a nonzero uncertainty in the momentumstarts with showing that the quantity
Is . Then using the differential operator form ofand integration by parts, show that it is also,
Together these show that if is. 0. then the preceding quantity must be 0. However, the Integral of the complex square of a function(the quantity in the brackets) can only be 0 if the function is identically 0, so the assertion is proved.
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