[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

consider particle of mass \(m\)
- with a spatially varying potential \(V(z)\) in the \(z\) direction
consider SE (schrodinger’s equation)
- \( -\frac{h^2}{2m}\frac{d^2 \psi(x)}{dz^2} + V(z) \psi(z) = E\psi(z) \)
- \(E\): energy of the particle
- \(\psi(z) \): 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
\( -\frac{h^2}{2m}\frac{d^2 \psi(x)}{dz^2} + V(z) \psi(z) = E\psi(z) \) becomes
- \( -\frac{h^2}{2m}\frac{d^2 \psi(x)}{dz^2} = E\psi(z) \) with boundary conditions
  - \( \psi(0) = 0 \)
  - \( \psi(L_z) = 0 \)
the general solution of this equation is of the form:
- \( \psi(z) = A \sin(kz) + B\cos(kz) \)
  - A and B are constants
  - \( k = \sqrt{\frac{2mE}{h^2}} \)
- \(B = 0 \) because \( \psi(0) = 0 \), as \(\cos(0) = 1\)
having simplified with one boundary condition applied
- \( \psi(z) = A \sin{(z)} \)
the condition \( \psi(L_z) = 0 \) means that
- \( kz \) must be a multiple of \( \pi \)
- \( k = \sqrt{\frac{2mE}{h^2}} = \frac{n\pi}{L_z} \)
  - where \( n \): integer
so, \( E = \frac{h^2k^2}{2m} \)
the solutions are
- \( \psi_n(z) = A_n \sin{\left(\frac{n\pi z}{L_z}\right)} \)
- \( E_n = \frac{h^2}{2m}\left(\frac{n\pi}{L_z}\right)^2 \)

particle in a box

\(n\) is only positive integers \( n = 1,2,… \), not zero
- \(\sin(a) = -\sin(a) \) 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
- \( \int_{0}^{L_z} \lvert A_n \rvert^2 \sin^2 \left( \frac{n\pi z}{L_z} \right) dz = \lvert A_n \rvert^2 \frac{L_z}{2} \)
- for this integral equal \(1\)
- choose \( \lvert A_n \rvert = \sqrt{\frac{2}{L_z}} \)
  - \(A_n\) can be complex
  - all such solutions are arbitrary within a unit complex factor
conventionally, we choose \(A_n\) real for simplicity in writing

particle-in-a-box

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

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

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

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