GR0177 #29



Alternate Solutions 
Qubbajsara 20161019 00:56:52  Of course if you sum the probability amplitude the answer will be one\\\\r\\\\nThe right answer is that the state with the highest probability is the one with eigen value rqual to 1 and probability amplitude equal to one half so its eigen value is the expectation value (most probable value)   davidkaleko 20100915 10:26:10  < O >  < + +  O  + + >$you can separate the above out to be 9 terms, withO> = 1*, O>=1*, O>=2*also, <  > = 1, while <  > = <  > = 0, due to orthogonality. This makes 6 of the 9 terms go to zero.plug it all in(1) + (1) + (2) = 1Sorry for the messy solution, I can't for the life of me figure out how to make a new line in this pseudolatex format!  

Comments 
camarasi 20170610 03:29:33  Definition of expectation value of an operator O is\r\n\r\n< O > = <O>\r\n\r\nPlugging in our state ...\r\n\r\n<O> = <++O(++)>\r\n\r\nNow we \'multiply out\' the inner product. This is similar to squaring a term like (a+b+c). You will get 9 terms  messy!\r\n\r\nBut remember that \r\n\r\n<> = \r\n\r\nwhere is the Kronecker delta function.\r\n\r\nSo only 3 terms survive.\r\n\r\n<O> = <O> + <O> + <O>\r\n\r\n=<O> + <O> + <O>\r\n\r\nProblem tells us that \r\n\r\n<O> = 1\r\n<O> = 1\r\n<O> = 2\r\n\r\nUsing these, \r\n\r\n<O> =  + + = 1\r\n\r\nAnswer: (C)\r\n   camarasi 20170610 03:07:40  Definition of expectation value of an operator O is \r\n\r\n <O>\r\n\r\n   Qubbajsara 20161019 00:56:52  Of course if you sum the probability amplitude the answer will be one\\\\r\\\\nThe right answer is that the state with the highest probability is the one with eigen value rqual to 1 and probability amplitude equal to one half so its eigen value is the expectation value (most probable value)   davidkaleko 20100915 10:26:10  < O >  < + +  O  + + >$you can separate the above out to be 9 terms, withO> = 1*, O>=1*, O>=2*also, <  > = 1, while <  > = <  > = 0, due to orthogonality. This makes 6 of the 9 terms go to zero.plug it all in(1) + (1) + (2) = 1Sorry for the messy solution, I can't for the life of me figure out how to make a new line in this pseudolatex format!   davidkaleko 20100915 10:20:04  < O >rnrn< + +  O  + + >rnyou can separate the above out to be 9 terms, withrnrnO> = 1*rnO>=1*rnO>=2*rnrnalso, <  > = 1, while <  > = <  > = 0, due to orthogonality. This makes 6 of the 9 terms go to zero.rnrnplug it all inrnrn(1) + (1) + (2) = 1   diane 20100105 20:42:25  1/6+1/2+2/3= (1+3+4)/6=8/6=4/3
so what gives?
Professor 20100112 11:55:53 
It is NOT 1/6, it is 1/6.

  lathena 20091008 21:31:41  Hi,
Further clarification on this problem would be nice. Why are the square rooted parts squared?
lathena 20091008 21:57:23 
err, ignore this. I thought about it more and it made sense.

buaasyh 20150825 11:55:46 
The coefficients of the orthogonal base states represent the probability amplitude to be in those states. The probability to be in those states are conjugate squares of the amplitudes. Because the amplitudes are real, the coefficients are squared to make the probabilities.

 

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