6. In lecture we considered the infinite square well located from 0< x < L. However, it is sometimes useful to consider the well to be centered at x = 0. Although this is really just a change in coordinates, it reveals additional properties of the system that are not as easily seen when the well is centered at x = L/2. So, we'll consider the infinite well centered at x = 0: U(x) = = Jo |x|< 1/1/2 √∞|x|> 1/1/201 a) Determine the stationary state wave functions and the energy spectrum using the same process as done in lecture (I recommend starting with sine and cosine functions). The following might be helpful: sin (-0) = -sin (0) and cos (-0) = cos(0). Show that the energy spectrum is the same as what we saw in lecture. For the stationary state wave functions, you should end up with just sine or just cosine depending on whether n is even or odd. b) Sketch the ground state wave function and the first three excited states (it doesn't need to be perfect, just a rough
6. In lecture we considered the infinite square well located from 0< x < L. However, it is sometimes useful to consider the well to be centered at x = 0. Although this is really just a change in coordinates, it reveals additional properties of the system that are not as easily seen when the well is centered at x = L/2. So, we'll consider the infinite well centered at x = 0: U(x) = = Jo |x|< 1/1/2 √∞|x|> 1/1/201 a) Determine the stationary state wave functions and the energy spectrum using the same process as done in lecture (I recommend starting with sine and cosine functions). The following might be helpful: sin (-0) = -sin (0) and cos (-0) = cos(0). Show that the energy spectrum is the same as what we saw in lecture. For the stationary state wave functions, you should end up with just sine or just cosine depending on whether n is even or odd. b) Sketch the ground state wave function and the first three excited states (it doesn't need to be perfect, just a rough
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Transcribed Image Text:6. In lecture we considered the infinite square well located from 0 < x < L. However, it is sometimes useful to consider
the well to be centered at x =0. Although this is really just a change in coordinates, it reveals additional properties of
the system that are not as easily seen when the well is centered at x = L/2. So, we'll consider the infinite well centered
at x = 0:
U (x)
So |x|< 1/1/2
{ ∞ |x|> 1/1/201
a) Determine the stationary state wave functions and the energy spectrum using the same process as done in lecture
(I recommend starting with sine and cosine functions). The following might be helpful: sin(-0) = sin (0) and
cos (-0) = cos (0). Show that the energy spectrum is the same as what we saw in lecture. For the stationary state
wave functions, you should end up with just sine or just cosine depending on whether n is even or odd.
b) Sketch the ground state wave function and the first three excited states (it doesn't need to be perfect, just a rough
sketch is fine). What do you notice about the shape of the wave function around x = 0 when n is even? When n is odd?
c) For what values of n is there always a node at the center of the well?
d) How do you think n is related to the number of anti-nodes?
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