Hello everyone,
I am reading about the Finite Square Well in Griffiths Quantum Mechanics Text. Right now, I am reading about the case in which the particle can be in bound states, implying that it has an energy E < 0. After some derivations, the author comes across the equation
[itex]\tan z = \sqrt{ \left(\frac{z_0}{z} \right)^2 - 1}[/itex]
where [itex]z = la[/itex] and [itex]z_0 = \frac{a}{\hbar} \sqrt{2mV_0}[/itex]; additionally, [itex]l= \frac{\sqrt{2m(E+V_0)}}{\hbar}[/itex].
The author makes the claim that, "if z_0 is very large, the intersections occur just slightly below [itex]z_n = n \frac{\pi}{2}[/itex], with n odd; it follows that
[itex]E_n + V_0 \approx \frac{n^2 \pi^2 \hbar^2}{2m(2a)^2}[/itex]" (1)
I don't quite see this. How does z_0 being large imply (1) is a true statement? I have tried to work out the details myself, but it appears that there are a lot of missing details which the author should have included; although, this is just my opinion.
I am reading about the Finite Square Well in Griffiths Quantum Mechanics Text. Right now, I am reading about the case in which the particle can be in bound states, implying that it has an energy E < 0. After some derivations, the author comes across the equation
[itex]\tan z = \sqrt{ \left(\frac{z_0}{z} \right)^2 - 1}[/itex]
where [itex]z = la[/itex] and [itex]z_0 = \frac{a}{\hbar} \sqrt{2mV_0}[/itex]; additionally, [itex]l= \frac{\sqrt{2m(E+V_0)}}{\hbar}[/itex].
The author makes the claim that, "if z_0 is very large, the intersections occur just slightly below [itex]z_n = n \frac{\pi}{2}[/itex], with n odd; it follows that
[itex]E_n + V_0 \approx \frac{n^2 \pi^2 \hbar^2}{2m(2a)^2}[/itex]" (1)
I don't quite see this. How does z_0 being large imply (1) is a true statement? I have tried to work out the details myself, but it appears that there are a lot of missing details which the author should have included; although, this is just my opinion.
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