Exercises on quantum mechanics

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   03:27, 2 Jul 2015

    
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The following are some problems that will help you refresh your memory about material you should have learned in undergraduate chemistry classes and that allow you to exercise the material taught in this text.

Chapters of Part 1

1.You should be able to set up and solve the one- and two-dimensional particle in a box Schrödinger equations. I suggest you now try this and make sure you see:

1. How the second order differential equations have two independent solutions, so the most general solution is a sum of these two.
2. How the two boundary conditions reduce the number of acceptable solutions from two to one and limit the values of E$E$ that can be “allowed”.
3. How the wave function is continuous even at the box boundaries, but dΨdx${\displaystyle \frac{d\mathrm{\Psi }}{dx}}$ is not. In general ddx${\displaystyle \frac{d}{dx}}$, which relates to the momentum because  −iℏddx$-i\hslash {\displaystyle \frac{d}{dx}}$ is the momentum operator, is continuous except at points where the potential V(x)$V(x)$ undergoes an infinite jump as it does at the box boundaries. The infinite jump in V$V$, when viewed classically, means that the particle would undergo an instantaneous reversal in momentum at this point, so its momentum would not be continuous. Of course, in any realistic system, V$V$ does not have infinite jumps, so momentum will vary smoothly and thus dΨdx${\displaystyle \frac{d\mathrm{\Psi }}{dx}}$ will be continuous.
4. How the energy levels grow with quantum number n$n$ as n2$n^2$ .
5. What the wave functions look like when plotted.

To continue reading click on the link below:

http://chem.libretexts.org/Textbook_Maps/Physical_and_Theoretical_Chemistry_Textbook_Maps/Map%3A_Advanced_Theoretical_Chemistry_(Simons)/Exercises