class35 - PHYS 5900 Class 35 (11/30/2009) Zi-Wei Lin Please...

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PHYS 5900 Class 35 (11/30/2009) Zi-Wei Lin Please fill out the Fall 2009 Online Student Opinion of Instruction Survey ( SOIS ) Survey Period: November 23 to midnight December 6 Repeat our re-definition of the ** operator: In[1]:= Unprotect @ NonCommutativeMultiply D ; In[2]:= A_ ** U : = A; U ** A_ : = A; In[4]:= A_ ** I B_ + C_ M : = A ** B + A ** C; I A_ + B_ M ** C_ : = A ** C + B ** C; In[5]:= NumberQ @ s D ^ = True; NumberQ @ t D ^ = True; In[6]:= scalarPatt2 = x_ ± ; Union @ Map @ NumberQ, Level @ x, 8 - 1 <DDD ± 8 True < ; In[7]:= A_ ** IH scalarPatt2 L * B_ M : = HH x L * A ** B L ; IH scalarPatt2 L * A_ M ** B_ : = HH x L * A ** B L ; In[9]:= Protect @ NonCommutativeMultiply D ; In[10]:= 8H 3 * A + B ± 5 L ** F, HH 2 * s ± Sqrt @ 5 * s * t DL * U L ** A ± s < Out[10]= : 3 A ** F + B ** F 5 , 2 A 5 s t > Now we can define the commutator [A,B]=AB-BA ( Note again: the non-commutative multiplication operator ** replaces the usual multiplication in the above but it is omitted for simplicity): In[11]:= comm @ A_, B_ D : = A ** B - B ** A

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We now prove some commutator identities: [A,B]=-[B,A] [s A,t B]=s t [A,B] [A,B+C]=[A,B]+[A,C] [A+B,C]=[A,C]+[B,C] [A,BC]=[A,B]C+B[A,C] [AB,C]=A[B,C]+[A,C]B [[A,B],C]+[C,A],B]+[B,C],A]=0 In[12]:= comm @ A, B D ± - comm @ B, A D Out[12]= True In[13]:= comm @ s * A, t * B D == s * t * comm @ A, B D ±± Simplify Out[13]= True In[14]:= comm @ A, B + C D == comm @ A, B D + comm @ A, C D Out[14]= True In[15]:= comm @ A + B, C D == comm @ A, C D + comm @ B, C D Out[15]= True In[16]:= comm @ A, B ** C D == comm @ A, B D ** C + B ** comm @ A, C D Out[16]= True In[17]:= comm @ A ** B, C D == A ** comm @ B, C D
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This note was uploaded on 04/25/2010 for the course PHYS 5900 taught by Professor Lin during the Fall '09 term at East Carolina University .

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class35 - PHYS 5900 Class 35 (11/30/2009) Zi-Wei Lin Please...

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