- #1
Lisa...
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Hi there!
I know how the wavefunctions look like for infinite potential wells and finite potential wells, with the barrier is placed at zero, so I got curious and drew the following well:
As you can see I split it into three different parts, because the Schrodinger equation has different solutions in each part.
To my opinion the wavefunction needs to equal:
In part 1: A sin(-k3) + B cos(-k3)= B cos(k3) - A sin(k3) (for all Energy levels)
In part 2: D e^(-ax) (for E1 and E2)
In part 3: F sin(k3) + G cos(k3) (for E3 till E5)
So the wavefunctions will look like sines in part 1 and 3 (starting at zero (at x=-3), because the barrier is infinite and ending at zero for the samen reason at x=3). In part 2 the function will be a decreasing exp-function, because of the U0 step in the well, which allows E1 and E2 to leak, though this function needs to be zero at x=3, because there is the infinite barrier.
Ok, so far so good, but how do I know the size of the amplitude for each energy level in each part of the well? And the frequency? (You don't need to explain this mathematically to me, just an explanation of how the frequencies and amplitudes of all energy levels are proportional to each other and why this is so will do ;-)! )
I know how the wavefunctions look like for infinite potential wells and finite potential wells, with the barrier is placed at zero, so I got curious and drew the following well:
As you can see I split it into three different parts, because the Schrodinger equation has different solutions in each part.
To my opinion the wavefunction needs to equal:
In part 1: A sin(-k3) + B cos(-k3)= B cos(k3) - A sin(k3) (for all Energy levels)
In part 2: D e^(-ax) (for E1 and E2)
In part 3: F sin(k3) + G cos(k3) (for E3 till E5)
So the wavefunctions will look like sines in part 1 and 3 (starting at zero (at x=-3), because the barrier is infinite and ending at zero for the samen reason at x=3). In part 2 the function will be a decreasing exp-function, because of the U0 step in the well, which allows E1 and E2 to leak, though this function needs to be zero at x=3, because there is the infinite barrier.
Ok, so far so good, but how do I know the size of the amplitude for each energy level in each part of the well? And the frequency? (You don't need to explain this mathematically to me, just an explanation of how the frequencies and amplitudes of all energy levels are proportional to each other and why this is so will do ;-)! )
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