Why are complex numbers used in the Schrodinger equation? Updated 2025-04-05 +Created 1970-01-01
Why is there a complex number in the equation? Intuitively and mathematically:
Necessity of complex numbers in the Schrödinger equation by Barton Zwiebach (2017)
Source. This useless video doesn't really explain anything, he just says "it's needed because the equation has an in it".
The real explanation is: they are not needed, they just allow us to write the equation in a shorter form, which is always a win: physics.stackexchange.com/questions/32422/qm-without-complex-numbers/557600#557600
Some good mentions at: Video "Where is Anatomy Encoded in Living Systems? by Michael Levin (2022)".
andor.oxinst.com/learning/view/article/measuring-resistance-of-a-superconducting-sample-with-a-dry-cryostat Not a video, but well done, by Oxford Instruments.
Superconductor, 4-probe measurement by Frederiksen Scientific A/S (2015)
Source. OK experiment, illustrates the educational kit they sell. No temperature control, just dumps liquid nitrogen into conductor and watches it drop. But not too bad either. The kit sale link is broken (obviously, enterprise stuff), but there are no archives unfortunately. But it must be some High-temperature superconductorGitHub account: github.com/hplgit
It should be mentioned that when you start Googling for PDE stuff, you will reach Han's writings a lot under his GitHub Pages: hplgit.github.io/, and he is one of the main authors of the FEniCS Project.
Unfortunately he died of cancer in 2016, shame, he seemed like a good educator.
He also published to GitHub pages with his own crazy markdown-like multi-output markup language: github.com/hplgit/doconce.
Rest in peace, Hans.
XML predecessor.
Once you hear about the uncomputability of such problems, it makes you see that all Diophantine equation questions risk being undecidable, though in some simpler cases we manage to come up with answers. The feeling is similar to watching people trying to solve the Halting problem, e.g. in the effort to determine BB(5).
The main way to sequence DNA methylation. Converts methylated cytosine to uracil, and then we can sequence those.
The key model database is located in the source code at
reconstruction/ecoli/flat.Let's try to understand some interesting looking, with a special focus on our understanding of the tiny E. Coli K-12 MG1655 operon thrLABC part of the metabolism, which we have well understood at Section "E. Coli K-12 MG1655 operon thrLABC".
We'll realize that a lot of data and IDs come from/match BioCyc quite closely.
reconstruction/ecoli/flat/compartments.tsvcontains cellular compartment information:"abbrev" "id" "n" "CCO-BAC-NUCLEOID" "j" "CCO-CELL-PROJECTION" "w" "CCO-CW-BAC-NEG" "c" "CCO-CYTOSOL" "e" "CCO-EXTRACELLULAR" "m" "CCO-MEMBRANE" "o" "CCO-OUTER-MEM" "p" "CCO-PERI-BAC" "l" "CCO-PILUS" "i" "CCO-PM-BAC-NEG"CCO: "Celular COmpartment"BAC-NUCLEOID: nucleoidCELL-PROJECTION: cell projectionCW-BAC-NEG: TODO confirm: cell wall (of a Gram-negative bacteria)CYTOSOL: cytosolEXTRACELLULAR: outside the cellMEMBRANE: cell membraneOUTER-MEM: bacterial outer membranePERI-BAC: periplasmPILUS: pilusPM-BAC-NEG: TODO: plasma membrane, but that is the same as cell membrane no?
reconstruction/ecoli/flat/promoters.tsvcontains promoter information. Simple file, sample lines:corresponds to E. Coli K-12 MG1655 promoter thrLp, which starts as position 148."position" "direction" "id" "name" 148 "+" "PM00249" "thrLp"reconstruction/ecoli/flat/proteins.tsvcontains protein information. Sample line corresponding to e. Coli K-12 MG1655 gene thrA:so we understand that:"aaCount" "name" "seq" "comments" "codingRnaSeq" "mw" "location" "rnaId" "id" "geneId" [91, 46, 38, 44, 12, 53, 30, 63, 14, 46, 89, 34, 23, 30, 29, 51, 34, 4, 20, 0, 69] "ThrA" "MRVL..." "Location information from Ecocyc dump." "AUGCGAGUGUUG..." [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 89103.51099999998, 0.0, 0.0, 0.0, 0.0] ["c"] "EG10998_RNA" "ASPKINIHOMOSERDEHYDROGI-MONOMER" "EG10998"aaCount: amino acid count, how many of each of the 20 proteinogenic amino acid are thereseq: full sequence, using the single letter abbreviation of the proteinogenic amino acidsmw; molecular weight? The 11 components appear to be given atreconstruction/ecoli/flat/scripts/unifyBulkFiles.py:so they simply classify the weight? Presumably this exists for complexes that have multiple classes?molecular_weight_keys = [ '23srRNA', '16srRNA', '5srRNA', 'tRNA', 'mRNA', 'miscRNA', 'protein', 'metabolite', 'water', 'DNA', 'RNA' # nonspecific RNA ]23srRNA,16srRNA,5srRNAare the three structural RNAs present in the ribosome: 23S ribosomal RNA, 16S ribosomal RNA, 5S ribosomal RNA, all others are obvious:- tRNA
- mRNA
- protein. This is the seventh class, and this enzyme only contains mass in this class as expected.
- metabolite
- water
- DNA
- RNA: TODO
rnavsmiscRNA
location: cell compartment where the protein is present,cdefined atreconstruction/ecoli/flat/compartments.tsvas cytoplasm, as expected for something that will make an amino acid
reconstruction/ecoli/flat/rnas.tsv: TODO vstranscriptionUnits.tsv. Sample lines:"halfLife" "name" "seq" "type" "modifiedForms" "monomerId" "comments" "mw" "location" "ntCount" "id" "geneId" "microarray expression" 174.0 "ThrA [RNA]" "AUGCGAGUGUUG..." "mRNA" [] "ASPKINIHOMOSERDEHYDROGI-MONOMER" "" [0.0, 0.0, 0.0, 0.0, 790935.00399999996, 0.0, 0.0, 0.0, 0.0, 0.0, 0.0] ["c"] [553, 615, 692, 603] "EG10998_RNA" "EG10998" 0.0005264904halfLife: half-lifemw: molecular weight, same as inreconstruction/ecoli/flat/proteins.tsv. This molecule only have weight in themRNAclass, as expected, as it just codes for a proteinlocation: same as inreconstruction/ecoli/flat/proteins.tsvntCount: nucleotide count for each of the ATGCmicroarray expression: presumably refers to DNA microarray for gene expression profiling, but what measure exactly?
reconstruction/ecoli/flat/sequence.fasta: FASTA DNA sequence, first two lines:>E. coli K-12 MG1655 U00096.2 (1 to 4639675 = 4639675 bp) AGCTTTTCATTCTGACTGCAACGGGCAATATGTCTCTGTGTGGATTAAAAAAAGAGTGTCTGATAGCAGCTTCTGreconstruction/ecoli/flat/transcriptionUnits.tsv: transcription units. We can observe for example the two different transcription units of the E. Coli K-12 MG1655 operon thrLABC in the lines:"expression_rate" "direction" "right" "terminator_id" "name" "promoter_id" "degradation_rate" "id" "gene_id" "left" 0.0 "f" 310 ["TERM0-1059"] "thrL" "PM00249" 0.198905992329492 "TU0-42486" ["EG11277"] 148 657.057317358791 "f" 5022 ["TERM_WC-2174"] "thrLABC" "PM00249" 0.231049060186648 "TU00178" ["EG10998", "EG10999", "EG11000", "EG11277"] 148promoter_id: matches promoter id inreconstruction/ecoli/flat/promoters.tsvgene_id: matches id inreconstruction/ecoli/flat/genes.tsvid: matches exactly those used in BioCyc, which is quite nice, might be more or less standardized:
reconstruction/ecoli/flat/genes.tsv"length" "name" "seq" "rnaId" "coordinate" "direction" "symbol" "type" "id" "monomerId" 66 "thr operon leader peptide" "ATGAAACGCATT..." "EG11277_RNA" 189 "+" "thrL" "mRNA" "EG11277" "EG11277-MONOMER" 2463 "ThrA" "ATGCGAGTGTTG" "EG10998_RNA" 336 "+" "thrA" "mRNA" "EG10998" "ASPKINIHOMOSERDEHYDROGI-MONOMER"reconstruction/ecoli/flat/metabolites.tsvcontains metabolite information. Sample lines:In the case of the enzyme thrA, one of the two reactions it catalyzes is "L-aspartate 4-semialdehyde" into "Homoserine"."id" "mw7.2" "location" "HOMO-SER" 119.12 ["n", "j", "w", "c", "e", "m", "o", "p", "l", "i"] "L-ASPARTATE-SEMIALDEHYDE" 117.104 ["n", "j", "w", "c", "e", "m", "o", "p", "l", "i"]Starting from the enzyme page: biocyc.org/gene?orgid=ECOLI&id=EG10998 we reach the reaction page: biocyc.org/ECOLI/NEW-IMAGE?type=REACTION&object=HOMOSERDEHYDROG-RXN which has reaction IDHOMOSERDEHYDROG-RXN, and that page which clarifies the IDs:so these are the compounds that we care about.- biocyc.org/compound?orgid=ECOLI&id=L-ASPARTATE-SEMIALDEHYDE: "L-aspartate 4-semialdehyde" has ID
L-ASPARTATE-SEMIALDEHYDE - biocyc.org/compound?orgid=ECOLI&id=HOMO-SER: "Homoserine" has ID
HOMO-SER
- biocyc.org/compound?orgid=ECOLI&id=L-ASPARTATE-SEMIALDEHYDE: "L-aspartate 4-semialdehyde" has ID
reconstruction/ecoli/flat/reactions.tsvcontains chemical reaction information. Sample lines:"reaction id" "stoichiometry" "is reversible" "catalyzed by" "HOMOSERDEHYDROG-RXN-HOMO-SER/NAD//L-ASPARTATE-SEMIALDEHYDE/NADH/PROTON.51." {"NADH[c]": -1, "PROTON[c]": -1, "HOMO-SER[c]": 1, "L-ASPARTATE-SEMIALDEHYDE[c]": -1, "NAD[c]": 1} false ["ASPKINIIHOMOSERDEHYDROGII-CPLX", "ASPKINIHOMOSERDEHYDROGI-CPLX"] "HOMOSERDEHYDROG-RXN-HOMO-SER/NADP//L-ASPARTATE-SEMIALDEHYDE/NADPH/PROTON.53." {"NADPH[c]": -1, "NADP[c]": 1, "PROTON[c]": -1, "L-ASPARTATE-SEMIALDEHYDE[c]": -1, "HOMO-SER[c]": 1 false ["ASPKINIIHOMOSERDEHYDROGII-CPLX", "ASPKINIHOMOSERDEHYDROGI-CPLX"]catalized by: here we seeASPKINIHOMOSERDEHYDROGI-CPLX, which we can guess is a protein complex made out ofASPKINIHOMOSERDEHYDROGI-MONOMER, which is the ID for thethrAwe care about! This is confirmed incomplexationReactions.tsv.
reconstruction/ecoli/flat/complexationReactions.tsvcontains information about chemical reactions that produce protein complexes:The"process" "stoichiometry" "id" "dir" "complexation" [ { "molecule": "ASPKINIHOMOSERDEHYDROGI-CPLX", "coeff": 1, "type": "proteincomplex", "location": "c", "form": "mature" }, { "molecule": "ASPKINIHOMOSERDEHYDROGI-MONOMER", "coeff": -4, "type": "proteinmonomer", "location": "c", "form": "mature" } ] "ASPKINIHOMOSERDEHYDROGI-CPLX_RXN" 1coeffis how many monomers need to get together for form the final complex. This can be seen from the Summary section of ecocyc.org/gene?orgid=ECOLI&id=ASPKINIHOMOSERDEHYDROGI-MONOMER:Fantastic literature summary! Can't find that in database form there however.Aspartate kinase I / homoserine dehydrogenase I comprises a dimer of ThrA dimers. Although the dimeric form is catalytically active, the binding equilibrium dramatically favors the tetrameric form. The aspartate kinase and homoserine dehydrogenase activities of each ThrA monomer are catalyzed by independent domains connected by a linker region.
reconstruction/ecoli/flat/proteinComplexes.tsvcontains protein complex information:"name" "comments" "mw" "location" "reactionId" "id" "aspartate kinase / homoserine dehydrogenase" "" [0.0, 0.0, 0.0, 0.0, 0.0, 0.0, 356414.04399999994, 0.0, 0.0, 0.0, 0.0] ["c"] "ASPKINIHOMOSERDEHYDROGI-CPLX_RXN" "ASPKINIHOMOSERDEHYDROGI-CPLX"reconstruction/ecoli/flat/protein_half_lives.tsvcontains the half-life of proteins. Very few proteins are listed however for some reason.reconstruction/ecoli/flat/tfIds.csv: transcription factors information:"TF" "geneId" "oneComponentId" "twoComponentId" "nonMetaboliteBindingId" "activeId" "notes" "arcA" "EG10061" "PHOSPHO-ARCA" "PHOSPHO-ARCA" "fnr" "EG10325" "FNR-4FE-4S-CPLX" "FNR-4FE-4S-CPLX" "dksA" "EG10230"
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