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| Keywords: Ensemble quantum 1DOF canonical.png en Ensemble canonically distributed over energy for a quantum system consisting of one particle in a potential well 2013-10-30 22 45 43 own Nanite Source Created with Matplotlib Python source code Requires matplotlib <source lang python > from pylab import figformat ' png' saveopts 'dpi' 300 'bbox_inches' 'tight' 'transparent' True 'frameon' True seterr divide 'ignore' Very important number smaller means more classical finer-spaced discrete levels larger means more quantum fewer discrete levels hbar 0 7/ 2 pi temp_canonical 4 1 energy_microcanonical -2 0 range_microcanonical 1 0 micro_e0 energy_microcanonical - 0 5 range_microcanonical micro_e1 energy_microcanonical + 0 5 range_microcanonical def potential x return x 6 + 4 x 3 - 5 x 2 - 4 x x linspace -2 5 2 5 1001 dx x1 - x0 U potential x mass 1 0 compute pixel edges used for pcolormesh xcorners zeros len x +1 xcorners len x x-0 5 dx xcorners-1 x-1 + 0 5 dx make an energy range for plots vs energy E linspace -20 20 10001 define color map that is transparent for low values and dark blue for high values weighted to show low probabilities well cdic 'red' 0 0 0 1 0 0 'green' 0 0 0 1 0 0 'blue' 0 0 7 0 7 1 0 7 0 7 'alpha' 0 0 0 0 1 0 4 0 4 0 2 0 6 0 6 0 4 0 8 0 8 0 6 0 9 0 9 1 1 1 cm_prob matplotlib colors LinearSegmentedColormap 'prob' cdic To get eigenvalues we need to set up a NxN matrix for the Schrodinger equation Hamiltonian For the momentum operator -hbar 2/ 2 m d 2/dx 2 the typical central difference approximation will be used H zeros len x len x set diagonal H ravel 0 len x +1 hbar hbar/ mass dx dx H ravel 0 len x +1 + U set above and below diagonal H ravel 1 len x +1 -0 5 hbar hbar/ mass dx dx H ravel len x len x +1 -0 5 hbar hbar/ mass dx dx Right the hamiltonian is set up so let's just go ahead and diagonalize it poink eigval eigvec eigh H def doev H Emax lowE_idx find eigval<Emax figure for i in lowE_idx plot x eigvec i label 'E '+str eigvali legend fontsize 8 micro eigval > micro_e0 eigval < micro_e1 1 0 print microcanonical E0 energy_microcanonical Delta 0 5 range_microcanonical avg energy print sum eigval micro /sum micro canonical exp -eigval/temp_canonical canonical_avgE sum eigval canonical /sum canonical print canonical T temp_canonical avg energy print canonical_avgE Boring level plot fig figure ax axes plot x potential x linewidth 3 for i in find eigval< 13 axhline eigvali color 0 5 0 5 0 5 linewidth 0 5 zorder -1 ylim -8 9 xlim -2 1 1 7 fig get_axes 0 xaxis set_ticks -2 -1 0 1 xlabel position x ylabel potential U x fig set_size_inches 3 3 fig patch set_alpha 0 savefig quant_potential_eigval_lines +figformat saveopts def levelplot weights Plot the potential with eigenstates' wavefunctions superimposed shown weights list fractions to multiply each eigenstate probability e g weight 0 do not show weight 1 fully show name filename to save to fig figure ax axes 0 08 0 1 0 73 0 89 0 125 0 1 0 71 0 8 plot x potential x linewidth 2 color 'r' zorder -1 maxp dx 3 5 amax weights eigwidth 0 2 for i in find eigval< 9 Here we plot the eigenfunctions as horizontal bars of varying darkness with height set by the energy eigenvalue if weightsi 0 continue don't plot levels with zero weight pdist eigvec i 2 weightsi pdist shape 1 len x extent amin x -0 5 dx amax x +0 5 dx eigvali-0 5 eigwidth eigvali+0 5 eigwidth img imshow vstack pdist pdist cmap cm_prob extent extent interpolation 'none' aspect 'auto' Alternate code using pcolormesh doesn't work because of ugly edges ycorners vstack eigvali-0 5 eigwidth len x +1 eigvali+0 5 eigwidth len x +1 pcolormesh vstack xcorners xcorners ycorners pdist cmap cm_prob clim 0 maxp ylim -9 9 xlim -2 1 1 7 fig get_axes 0 xaxis set_ticks -2 -1 0 1 ax xaxis set_ticklabels ax yaxis set_ticklabels ax xaxis labelpad 2 ax yaxis labelpad -3 xlabel position x ylabel energy ax axes 0 83 0 1 0 14 0 89 axisbg 0 95 0 95 0 95 ax xaxis set_ticks ax yaxis set_ticklabels ax yaxis set_ticks_position 'right' ylim -9 9 xlabel states dos E 0 0 for i Elevel in enumerate eigval Here we sum up the density of states function if Elevel > 20 continue don't waste time with high levels dos + exp -4 E-Elevel /eigwidth 2 weightsi fill_betweenx E dos linewidth 0 color 0 2 0 2 0 76 xlim -0 05 max dos max dos 1 1 fig set_size_inches 3 3 fig patch set_alpha 0 levelplot ones len eigval savefig quant_potential_eigval_pdists +figformat saveopts levelplot micro sca gcf axes0 axhspan micro_e0 micro_e1 color 0 7 1 0 7 zorder -2 sca gcf axes1 axhspan micro_e0 micro_e1 color 0 7 1 0 7 zorder -2 savefig quant_potential_eigval_pdists_micro +figformat saveopts levelplot canonical sca gcf axes0 annotate \\langle E\\rangle -0 5 canonical_avgE textcoords None verticalalignment 'top' color 0 0 4 0 axhline canonical_avgE linestyle 'dotted' linewidth 1 color 0 0 4 0 annotate 1 2 7 -temp_canonical 1 2 7 arrowprops 'arrowstyle' '<->' text 1 15 7 -0 5 temp_canonical ' kT ' horizontalalignment 'right' verticalalignment 'center' sca gcf axes1 axhline canonical_avgE linestyle 'dotted' linewidth 1 color 0 0 4 0 fill_betweenx E exp -E/temp_canonical linewidth 0 color 0 7 1 0 7 zorder -2 green exponential savefig quant_potential_eigval_pdists_canonical +figformat saveopts Position expectation values figure pdist zeros len x for i p in enumerate micro pdist + p eigvec i 2 if any micro plot x pdist/sum micro /dx label 'microcanonical' pdist zeros len x for i p in enumerate canonical pdist + p eigvec i 2 plot x pdist/sum canonical /dx label 'canonical' color 'g' xlim -2 1 1 7 fig get_axes 0 xaxis set_ticks -2 -1 0 1 xlabel position x ylabel PDF of position P x legend savefig quant_position_pdf +figformat saveopts </source> cc-zero Images with Python source code Uploaded with UploadWizard 1D potential wells Statistical ensemble | ||||
