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# Quantum Mechanics I PHY 851

MSU

GPA 3.67

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This 2 page Class Notes was uploaded by Quinn Larkin on Saturday September 19, 2015. The Class Notes belongs to PHY 851 at Michigan State University taught by Staff in Fall. Since its upload, it has received 24 views. For similar materials see /class/207632/phy-851-michigan-state-university in Physics 2 at Michigan State University.

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Date Created: 09/19/15

Lecture 6 February 7 2001 1 The Harmonic Oscillator One of the most important problems in physics is the harmonic oscillator In fact quantum eld theory considers every point in space to have it s own oscillator We will start with a single oscillator with the Hamiltonian 2 2 p 7m 2 H 7 L39 1 2m 2 where the spring constant is expressed in terms of a frequency Here L39 and p are operators Dirac s solution to the problem is to de ne two new operators mu 1 f mw 1 Ediv 17 7w 17 2 Qh LH 2hmwp 2M 27ame U The operators are known the annihilation or destruction and creation operators respec tively for reasons to be seen below The operators satisfy simple commutation relations la aTl 11 3 which can be checked by substituting the expressions for L39 and 1 into the commutator Furthermore the Hamiltonian may be written H hw llal 4 To see that the creation operator does exactly what is sounds like consider an eigenstate of the Hamiltonian such that 5 By using the commutation relations one can see that aaltangtgt n 1gtltangtgt 6 by commuting the al to the far left Performing the same trick with the state one nds raw n 1gtltangtgt lt7 The operator alt is referred to the number operator and since the energy is expressed in terms of the number operator and since that energy must not have negative values one of the eigenstates must have eigenvalue n 0 Otherwise one could lower the number and therefore the energy to an arbitrarily low number by succesive operations of 1 Thus one knows that eigenstates of the number operator are 0124 4 A but one does not yet know the normalization of the states Lecture 6 February 7 2001 To see the effect of the creation operators toward normalization one considers the norm of the state alm where m is a normalized state ngaalm 71 1 8 where one has commuted al past 1 Therefore one can see that the creation and destruction operators have the following effect on the normalization film V71 1m 1 am Wm 1 9 Note that the destruction operator returns zero when acting on the ground state where 71 0 We will see later on that similar tricks are used with angular momentum raising and lowering operators Finally it is straightforward to find the ground state wave function if one is clever enough to guess that the form of the solution is a Gaussian W 10 For the moment we neglect the normalization To show that it is a solution we first take derivatives with respect to dixWDW w037a d72le 2737 QTYMSW 11 Plugging this into the Schrodinger equation 72 32 1 mw 7 7 7 r2 7Er2 12 2m 14 1 WWW 2 WW WCquot allows one to determine a and E by inspection 7 1 a 7 E 772w 13 mw 2 One may also calculate the normalization Z by enforcing the constraint 2 2 2 0C 3 3 ZlAE m 2a 1 ding lt5 a39 X This gives Z 17r14a12 If one were to consider an 7 dimensional problem 2 2 2 pr 2 2 2 mfx ixll 2 2 2 mixing 1a one can write the solution as 1 1232 53 11W22 WASH 16 where 13 are the solutions to the 1d Schrodinger equation and E 17 Examples

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