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Dif­fer­en­tial Equa­tions are the one of the primary math­em­at­ic­al tools used in sci­ence and engin­eer­ing. Phys­ics is abso­lutely no excep­tion and it’s almost impossible to per­form any mean­ing­ful phys­ics without util­ising dif­fer­en­tial equa­tions. At their heart is the descrip­tion of both the time and spa­tial vari­ations of a par­tic­u­lar quant­ity. The clas­sic example of a dif­fer­en­tial equa­tions is the ‘wave equa­tion’. This equa­tion (for example) is seen in the study of springs, sound, light, heat, elec­tro­mag­net­ism and many many more! The solu­tions of the equa­tions are the ‘wave­func­tions’; these are the phys­ic­ally sig­ni­fic­ant quant­it­ies which describe the motion etc., of particles. In this playl­ist I dis­cuss the meth­od of power series solu­tions, the meth­od of Frobeni­us, Laplace’s Equa­tion, the Sch­rodinger Equa­tion and finally I derive the wave­func­tions of the hydro­gen atom (this clearly requires some phys­ics also).

Return to Video Tutori­als A-F

Power Series Solu­tions to DEs

Meth­od of Frobenius

The Laplace Equation

The Meth­od of Sep­ar­a­tion of Variables

The Wave­func­tions of the Hydro­gen Atom

Video # Video Tutori­al Title Remarks
1 Clas­si­fic­a­tion of Dif­fer­en­tial Equations Dis­cuss­ing all the terms by which we clas­si­fy dif­fer­en­tial equations
2 What are Power Series Used For? I try to motiv­ate the prac­tic­al use of power series in physics
3 Where Power Series Start Do they start at n=0, n=1, n=3 etc?
4 Shift­ing Power Series Indices A very import­ant skill; shift­ing from n=1 to n=0 (say)
5 Dif­fer­en­ti­at­ing Power Series As simple as it sounds but import­ant when we dis­cuss the meth­od of Frobenius
6 Mul­tiply­ing Power Series by Functions Dis­cuss­ing the impact on where the power series start
7 The Char­ac­ter­ist­ic Equa­tion 1/2 Deriv­ing the Char­ac­ter­ist­ic Equa­tion for solv­ing equa­tions with con­stant coefficients
8 Intro­duc­tion to Power Series Solutions How dif­fer­en­tial equa­tions are solved using the meth­od of power series solutions
9 Gen­er­al and Par­tic­u­lar Solutions Not yet recorded
10 The Gen­er­al Solution To 2nd order D.Es., with con­stant coef­fi­cients using power series
11 Power Series Example 1 y”- y’ = 0
12 Power Series Example 2 y” + 4y = 0
13 Power Series Example 3 y” — 9y = 0
14 a Proof of the Wronskian Find out if your solu­tions are lin­early depend­ent or not
14 b Intro­duc­tion to the Meth­od of Frobenius Solv­ing equa­tions which are not ana­lyt­ic using power series
15 a Example 1 Part 1 x2 + y” + (x — x2)y’ — y =0
15 b Example 2 Part 2 x2 + y” + (x — x2)y’ — y =0
16 a Example 1 Part 1 xy” + 2y’ + xy = 0
16 b Example 1 Part 2 xy” + 2y’ + xy = 0
17 Example 3 (x2 + 1)y” + xy’ — y = 0
18    
19    
20 Intro­duc­tion to Laplace’s Equation V” = 0
21 Earnshaw’s The­or­em No loc­al max­ima or min­ima are permitted
22 First Unique­ness Theorem Why the function’s value at the bound­ary is important
23 Second Unique­ness Theorem  
24 Intro­duc­tion to Sep­ar­a­tion of Variables The fun­da­ment­als of using this very power­ful tech­nique to solve dif­fer­en­tial equations
25 Char­ac­ter­ist­ic Equa­tion 2/2 Deriv­a­tion of the Char­ac­ter­ist­ic Equation
26 Sep­ar­a­tion of Vari­ables Example 1  V” = 0 to include the use of Four­i­er Series
27 Sep­ar­a­tion of Vari­ables Example 2  V” = 0 (new bound­ary con­di­tions) to include the use of Four­i­er Series
28 Sep­ar­a­tion of Vari­ables Example 3  V” = 0 (new bound­ary con­di­tions) to include the use of Four­i­er Series
29 Prop­er­ties of Sep­ar­able Solutions  Why sta­tion­ary state solu­tions are the fun­da­ment­als of Quantum Mechanics
30 Spher­ic­al Polar Co-ordinates  Rect­an­gu­lar to spher­ic­al polar co-ordin­ates is required for the wave­func­tions of Hydrogen
31 Laplace Equa­tion in Spher­ic­al Co-ords Con­vert­ing the D.E., to spher­ic­al polar coordinates
32 The Radi­al Equation The first of the three ordin­ary equa­tions res­ult­ing from the meth­od of sep­ar­a­tion of variables
33 Lapla­cian in Spher­ic­al Coordinates A very tedi­ous deriv­a­tion of a very import­ant operator
34 The Azi­muth­al Equation The second of the three ordin­ary equa­tions res­ult­ing from the meth­od of sep­ar­a­tion of variables
35 The Polar Angle Equation The third of the three ordin­ary equa­tions res­ult­ing from the meth­od of sep­ar­a­tion of variables
36 Sch­rodinger Equa­tion in Spher­ic­al Co-ords Sim­pli­fy­ing the deriv­a­tion of the Hydro­gen atom wavefunctions
37 The Radi­al Equa­tion for Hydrogen  
38 Wave­func­tions of Hydrogen A deriv­a­tion of the wave­func­tions of the Hydro­gen atom
39 Quantum Num­bers Where they come from and why they are useful
40 Elec­tron Orbitals The Peri­od­ic Table

 

dif­fer­en­tial equa­tions frobeni­us laplace’s equa­tion vari­ables sep­ar­a­tion com­plete­ness ortho­gon­al ortho­gon­al­ity dif­fer­en­tial equa­tions frobeni­us laplace’s equa­tion vari­ables sep­ar­a­tion com­plete­ness ortho­gon­al ortho­gon­al­ity dif­fer­en­tial equa­tions frobeni­us laplace’s equa­tion vari­ables sep­ar­a­tion com­plete­ness ortho­gon­al orthogonality 

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