Mechanica Quantica En Ingles

Lecture 1

Einstein and de broglie relations:
E=ℏω
p=ℏk
kl is the phase change. Where l is the distance.
When propagating EM waves behave like waves when measured like particles.
Hamiltonian mechanics and poisson brackets. Used to show mechanics not so different from quantum mechanics after all.
Lecture 2.
Trying to get a wave equation for matter from similarities with the proper waveequation.
Has to satisfy the energy relationship for kinetic energy which is
E=p22m not including the potential energy (free particle)
Substituting we get the dispersion relation for matter waves
ω=ℏk22m
The power of each indicates how many derivatives we take so k has second omega has first.
We see cant use sin or cosine functions, use exponential instead.
eikx-ωt
We write down theschrodinger equation in the form
iℏ∂Ψ∂t=-ℏ22m∂2Ψ∂x2
Because ∂2Ψ∂x2=-k2Ψ so after substituiting to equation we get energy
Wavefunction, generally complex square of its modulus gives probability density. Integrating between initial point and final gives probability of finding within the point. Integral over all possible places always has to give one (has to be somewhere).
Writing down thesolution as real and imaginary part and subbing into tdse shows its not too different from Hamiltonian classical mechanics.

Lecture 3.
We intend to solve the schrodinger equation by separation of variables Ψ=uxT(t)
Sub into equation and make sure there is one of each variables at each side of equation.
This implies both sides for set values of each variable are equal to a constant. If you set x,the side with t is equal to constant too and vice versa.

Now solve for both sides separately and see what happens. For the time dependant LHS simple integration to get
T=T0e-iEt/ℏ
Total solution includes spatial part
ψ=u(x) e-iEt/ℏ
When we take the modulus squared of that we see that it only yields ux^2 so the probability for separation of variables solutions does not change with time.(stationary state). Only if one value of energy
From the other part of the equation we obtain the time independent schrodinger equation. Just multiply through by u(x) and you got it. This cannot be solved like the time dependence one, we need a given value of potential first (particular case).
For a free particle, solutions are the same as for SHO so sines and cosines. [Sin(kx)]
Generalsolution written as u=Asinkx+Bcos⁡(kx) equivalently in exponential form
u=Ceikx+De-ikx
Hence total solution is both spatial and temporal together giving ψ=ψ0e-iEt/ℏeikx yielding
ψ=ψ0e-iωt-kx
Which again gives contant probability in time in case of one energy.
Superposition occurs due to the linearity of Schrodinger equation and is the only way of getting probability density varying with time.This happens when we have more than one solution, hence more than one value of energy allowed (as in free particles). The exponential terms do not cancel out in this case. However if the energies happen to be equal to each other for different solutions again we get constant probability.

Lecture 4.
Infinite square well.
Within the well situation very much alike free particle only thatboundary conditions act. Set boundar conditions and subtract the two equations or add getting even and odd parity. Since its 2Bsin(ka)=0 solve this for for appropriate ka and you get the allowed values of k. Another constraint is the normalisation. Proability within the well is one. So integral of modulus squared within the well must give 1.
For energy sub the value of k in terms of all the othercoefficients into the de Broglie dipersion relation for energy.
Parity is the symmetry of the solutions withing the well. If function symmetric (cosine) even parity if antisymmetric (sine) odd parity.
Only superimposing solutions gives time dependant changing probability.
TISE solution: if E>V, k=2m(E-V)ℏ sinces and cosines solution.
E<V exponential solutions, γ=2m[V-E]ℏ=ik u=Ce±γx...