[QMSE] W02 - Solving for particle in a box

• simplest problem in quantum mechanics
  • useful to show the main features of a whole class of problems called the energy eigen problems
  • introduces the idea of quantization and energy eigenvalues with their eigen functions

problem description

• consider particle of mass $m$
  • with a spatially varying potential $V\left(z\right)$ in the $z$ direction
• consider SE (schrodinger’s equation)
  • $-\frac{h^2}{2m}\frac{d^2\psi \left(x\right)}{d{z}^2}+V\left(z\right)\psi \left(z\right)=E\psi \left(z\right)$
  • $E$: energy of the particle
  • $\psi \left(z\right)$: is its wavefunction
• assume the potential energy is a simple rectangular potential well
  • higher up the well is higher energy
  • horizontally the position of the particle is described
• characteristics of the potential well
  • $L_z$: the thickness of the well
  • $V=0$: potential energy is $0$ at the bottom
  • rising up to $\infty$ vertically at the walls $z=0$ and $z=L_z$
    • so this is an infinite or infinitely deep potential well
• because potentials at $z=0$ and $z=L_z$ are infinitely high
  • but particle’s energy, $E$, is finite
  • the assumption is that there is no possibility of finding the the particle outside the walls of the well
    • i.e. beyond $z<0$ and $z>L_z$
  • so the wavefunction $\psi$ at and outside the walls is $0$

SE inside the well

• inside the well with the above specifications,
  • SE simplifies a bit
• $-\frac{h^2}{2m}\frac{d^2\psi \left(x\right)}{d{z}^2}+V\left(z\right)\psi \left(z\right)=E\psi \left(z\right)$ becomes
  • $-\frac{h^2}{2m}\frac{d^2\psi \left(x\right)}{d{z}^2}=E\psi \left(z\right)$ with boundary conditions
    • $\psi \left(0\right)=0$
    • $\psi \left(L_z\right)=0$
• the general solution of this equation is of the form:
  • $\psi \left(z\right)=A\sin \left(kz\right)+B\cos \left(kz\right)$
    • A and B are constants
    • $k=\sqrt{\frac{2mE}{h^2}}$
  • $B=0$ because $\psi \left(0\right)=0$, as $\cos \left(0\right)=1$
• having simplified with one boundary condition applied
  • $\psi \left(z\right)=A\sin \left(z\right)$
• the condition $\psi \left(L_z\right)=0$ means that
  • $kz$ must be a multiple of $\pi$
  • $k=\sqrt{\frac{2mE}{h^2}}=\frac{n\pi }{L_z}$
    • where $n$: integer
• so, $E=\frac{h^2{k}^2}{2m}$
• the solutions are
  • ${\psi }_n\left(z\right)=A_n\sin \left(\frac{n\pi z}{L_z}\right)$
  • $E_n=\frac{h^2}{2m}{\left(\frac{n\pi }{L_z}\right)}^2$

particle in a box

• $n$ is only positive integers $n=1,2,\dots$, not zero
  • $\sin \left(a\right)=-\sin \left(a\right)$ for any real number $a$
• the wavefunctions with negative $n$ are the same as those with positive $n$
  • with an arbitrary factor of $-1$
• for $n=0$, the wavefunction is $0$ everywhere
• normalizing the wavefunctions
  • ${\int }_0^{L_z}|A_n{|}^2{\sin }^2\left(\frac{n\pi z}{L_z}\right)dz=|A_n{|}^2\frac{L_z}{2}$
  • for this integral equal $1$
  • choose $\left|A_n\right|=\sqrt{\frac{2}{L_z}}$
    • $A_n$ can be complex
    • all such solutions are arbitrary within a unit complex factor
• conventionally, we choose $A_n$ real for simplicity in writing

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eigenvalues and eigenfunctions

• only solutions with a specific set of allowed values of a parameter (like energy)
  • those values are eigenvalues
• each eigenvalue has a particular wavefunction
  • those are eigenfunctions
• eigenvalues and eigenfunctions are together termed eigensolutions
• since the parameter here is energy
  • eigenvalues are called eigenenergies
  • eigenfunctions are called energy eigenfunctions

degeneracy

• it is possible to have one that one eigenfunction for the same eigenvalue
  • this phenomenon is known as the degeneracy of that particular eigenvalue

parity of wavefunctions

• eigenfunctions have definite symmetry
• even parity
  • when function is mirror image on hte left of what it is on the right
  • also called even function
  • $n=1,3$
• odd parity
  • inverted image on the other half side
  • the value on the other half is minus times given value on this half
  • also called odd function
• eigenfunctions in the infinite well alternate between even and odd
  • all the solutions have definite parity
• definite parity is common in symmetric problems
  • helpful mathematically
  • lot of integrals vanish out because of this
  • emergence of quantum behavior
• electrons behave like propagating waves
  • mono energetic waves
• put particle in box
  • electron has only discrete states of energy with specific energy eigenfunctions and eigenenergies
  • this is the quantum behavior of particles

quantum confinement

• differences from classical (CM) paradigm
  1. only a discrete set of possible values for the energy
  2. there is a minimum possible energy of the particle corresponding to $n=1$
     • here $E_1=\left(\frac{h^2}{2m}\right){\left(\frac{\pi }{L_z}\right)}^2$
     • sometimes called a zero-pint energy state exists as $L_z\to \infty$
     • because of confinement of system to small dimensions
  3. the particle is not uniformly distributed over the box
     • its distribution is different for different energies
     • it is almost never found very near to the box walls
     • the probability obeys a standing wave pattern
     • these aspects are nothing like CM
     • the eigenfunctions indicate the probability of finding the particle at the given eigenenergy
     • each successive higher energy state has one more zero probability point along the wave function

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energies in quantum mechanics

• joules can always be used as energy units
  • but these are large
• a more convenient energy unit: electron-volt (eV)
  • $1$ electron-volt $eV$ $\simeq 1.692\times {10}^{-19}$
• $1eV$ is the energy change of an electron in moving through an electrostatic potential change of $1V$
  • Energy in $eV=$ energy in $\frac{Joules}{electronic-charge\left(e\right)}$
  • electronic charge $=1.602176565\times {10}^{-19}C$ (Coulumbs)

order of magnitude

• example:
  • confine an electron in a $5\AA \left(0.5nm\right)$ thick
    • about the size of a hydrogen atom
  • $E_1=\left(\frac{h^2}{2m_0}\right){\left(\frac{\pi }{5\times {10}^{-10}}\right)}^2\cong 2.4\times {10}^{-19}J\cong 1.5eV$
  • the separation between the first and second allowed energies: $(E_2-E_1\simeq 3E_1)\simeq 4.5eV$
    • this is the characteristic size of major energy separations between levels in an atom
  • while the particle in a box is not an accurate model of an atom, it gives the magnitudes of energy for electrons in an atom