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The Quantum Lin­ear Har­mon­ic Oscil­lat­or (LHO) is one of the first applic­a­tions of quantum mech­an­ics to the real world and is gen­er­ally one of the first dif­fi­cult top­ics stud­ied when stu­dents begin quantum mech­an­ics. While it may seem at a first glance to be merely a text­book example; the lin­ear har­mon­ic oscil­lat­or poten­tial is found all over phys­ics and engin­eer­ing. For this reas­on, in order to build new devices etc., we require often genu­inely require the LHO solu­tions to the Sch­rodinger Equa­tion. The two meth­ods for deriv­ing the wave­func­tions are the ana­lyt­ic (brute force) meth­od and the Lad­der Oper­at­or (nifty) meth­od. The videos I present below dis­cuss the Lad­der Oper­at­or meth­od; these are very use­ful at a later stage dur­ing the study of angu­lar momentum (orbit­al and spin). Per­haps in the future I’ll get around to record­ing videos dis­cuss­ing the ana­lyt­ic method.

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Video # Video Tutori­al Title Remarks
1 Dimen­sion­less Sch­rodinger Equation Algeb­ra­ic Meth­od : Re-write the Sch­rodinger Equa­tion in terms of dimen­sion­less quantities
2 Hamilto­ni­an Using Lad­der Operators Re-write the Hamilto­ni­an Oper­at­or in terms of Lad­der Oper­at­ors (y-d/dy) & (y+d/dy)
3 Cre­ation / Rais­ing Operator (y — d/dy) Raises the wavefunction’s energy by one unit
4 Anni­hil­a­tion / Lower­ing Operator (y + d/dy) Lowers the wavefunction’s energy by one unit
5 Lad­der Oper­at­ors : Pos­i­tion and Momentum Re-write the Lad­der Oper­at­ors in terms of both pos­i­tion and momentuma+- = xˆ +- i pˆ
6 Apply­ing Lad­der Operators Quite a subtle argument
7 Pos­i­tion and Momentum as Lad­der Operators Remov­ing the (y +- d/dy) and util­ising only pos­i­tion ‘x’ and momentum ‘P’
8 The Expect­a­tion Value of Position The value obtained upon meas­ure­ment of particles pre­pared similarly
9 The Square of the Pos­i­tion Operator The oper­at­or obtained when we square the Pos­i­tion Operator
10 The Expect­a­tion Value of Pos­i­tion Squared The value obtained upon meas­ure­ment of particles pre­pared similarly
11 The Ground State Wavefunction The low­est energy Quantum LHO wave­func­tion obtainable
12 Nor­m­al­ising the Ground State Wavefunction Nor­m­al­ising the expect­a­tion value of the wave­func­tion to 1 over all space
13 High­er Order Wavefunctions Gen­er­at­ing wave­func­tions above the ground state using Lad­der Operators
14 Nor­m­al­isa­tion of the Second Order Wavefunction A deriv­a­tion which is used later to derive the occu­pancy function
15 Nor­m­al­ised Wavefunctions Ana­lys­ing the nor­m­al­isa­tion of the 1st & 2nd wave­func­tions IOT nor­m­al­ise all fur­ther wavefunctions
16 Third Order Wavefunction The second excited wavefunction
17 Zero Point Energy The energy asso­ci­ated with the ground state wave­func­tion (low­est pos­sible LHO energy)
18 Zero Point Energy Using Lad­der Operators One of the most import­ant the­or­ems from clas­sic­al mech­an­ics! n units to n+1 or n-1 units
19 Expect­a­tion of Kin­et­ic Energy Operator Deriv­a­tion of the K.E., Oper­at­or and its asso­ci­ated expect­a­tion value
20 Expect­a­tion of the Poten­tial Energy Operator Deriv­a­tion of the P.E., Oper­at­or and its asso­ci­ated expect­a­tion value

 

 

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