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Problem%20Set%201%202008.1

# Problem%20Set%201%202008.1 - 3 An operator A Î which...

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Due date Mar. 19 Wed. 12:00 pm 1. Sakurai Chap.1, 2 2. Here are some numerical values: ħ =1.054887x10 -27 erg-sec. Charge of proton=1.60218x10 -19 coulombs speed of light, c=2.99792458x10 10 cm/s (a) A fermi is 10 -13 cm, and is a widely used unit in nuclear and high-energy physics. Derive the value of ħ c in MeV-fermi. (b) If charge is measured in coulombs, the electrostatic energy of two electrons a distance r meters apart is V=e 2 /4 πε 0 . Since ħ c has dimension of energy-distance, one can write V= αħ c/r where α is dimensionless and therefore thankfully independent of unit conventions. Calculate the value of α from e in coulombs and ε 0 in MKS units.
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Unformatted text preview: 3. An operator A( Î» ) , which depends explicitly on a continuous parameter , is said to be differentiable if d A d A d dA d ) ( ) ( lim âˆ’ + = âˆž â†’ exists. (a) if two operators A and B are differentiable, show that Î»Î» d dB A B d dA AB d d + = ) ( (b) if A is differentiable and has its inverse, show that 1 1 1 âˆ’ âˆ’ âˆ’ âˆ’ = A d dA A d dA 4. For any operators A and B, and for a real parameter , prove that exp( Î» B)A exp(-Î» B)=A+i Î» [B, A] - Î» 2 /2[B,[B,A]]+â€¦ , where the nth term is ]]]] , [ , [ , [ , [ ! 1 A B B B B i n n n L . This is the Baker-Hausdorff lemma....
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