[QMSE] W02 - Particle in a box

particle in a box problem

• the schrodinger’s equation (SE) is used for discrete states at very specific energy levels

---

linearity and normalization

• schrodinger’s equation is linear
• the wavefunction $\psi$ appears only in first order
  • there is no second or higher order terms
  • such as ${\psi }^2$ or ${\psi }^3$
• so, if $\psi$ is a solution, so is $a\cdot \psi$
  • regardless of the constant $a$ value
• quantum mechanics linearity is very general
  • QM equations are linear in the QM amplitude for which the equation is being solved
• unlike CM, this is an absolute property in QM
• allows use of linear algebra for QM math

normalization

• since SE is linear, the wave function can be normalized

• born’s postulate says that the probability of finding a particle near some point vector $\text{r}$
  • proportional to the modulus squared of the wave function
  • mathematically, $P\left(\text{r}\right)\propto \left|\psi \right(\text{r}){|}^2$
• assume $P\left(\text{r}\right)$ to be a probability density
  • for some infinitesimal volume $d^3\text{r}$ around $\text{r}$
• the probability of finding the particle in that volume is $P\left(\text{r}\right)d^3\text{r}$
• the sum of all such probabilities should be 1
  • $\int P\left(\text{r}\right)d^3\text{r}=1$
• this integral usually gives some other real positive number
  • this necessitates normalization
• consider the integral output to be
  • $\frac{1}{|a{|}^2}$
  • $a$ is possibly complex

• so $\int \left|\psi \right(\text{r}){|}^2{d}^3\text{r}=\frac{1}{|a{|}^2}$

• we know that $\psi \left(\text{r}\right)$ is a solution to SE
  • so even $a\psi \left(\text{r}\right)$ is a solution
  • consequence of linearity of SE
• so, assume a solution ${\psi }_N=a\psi$ instead of $\psi$
  • then $\int \left|{\psi }_N\right(\text{r}){|}^2{d}^3\text{r}=1$
• we can use $\int \left|{\psi }_N\right(\text{r}){|}^2$ as the probability density
  • $P\left(\text{r}\right)=\int \left|{\psi }_N\right(\text{r}){|}^2$
  • this is the “normalized wavefunction”

steps to normalize

• take the solution $\psi$ obtained from SE
• integrate $\left|\psi \right(\text{r}){|}^2$ to get $\frac{1}{|a{|}^2}$
• find $a$ to obtain ${\psi }_N=a\psi$
  • for which $\int \left|{\psi }_N\right(\text{r}){|}^2{d}^3\text{r}=1$
• the new probability density is $\left|{\psi }_n\right(\text{r}){|}^2$

note

• normalization only sets the magnitude of $a$
  • not the phase
  • free to choose any phase for $a$
• if space is infinite, $\sin \left(kx\right),\cos \left(kz\right)and\exp (i\text{k}\cdot \text{r})$ cannot be normalized this way
  • their squared modulus is not “Lebesgue integrable” i.e. not L2 functions
  • result of over-idealizing the math to get functions that are simple to use
    • this is a mathematical difficulty, not physical
• workarounds:
  1. work in finite volumes in actual problems
  2. use normalization to a delta function introduces another infinity to compensate for the first one