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p827aps3

# p827aps3 - p i = ∂ L/∂ ˙ x i Show that if L = T − V...

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Physics 827: Problem Set 3 Due Wednesday, October 10 by 11:59 PM Each problem is worth 10 points. 1. Shankar, problem 1.9.2. 2. Shankar, problem 1.9.3. 3. Shankar, problem 1.10.3. 4. Consider the operator T = CK 2 , where K = iD and D is the deriva- tive operator in one dimension, as discussed in class (see Shankar, pp. 63-67), and C is a real scalar constant. Assume that T acts on the entire real axis, −∞ < x < . (a). Show that T is Hermitian. (b). Find the eigenvalues and eigenvectors of T . Write the eigenvalues as Ck 2 and the corresponding eigenvectors as | k ) , so that T | k ) = Ck 2 | t ) . What are the eigenvectors ( x | k ) in the x basis? Choose the normalization ( k | k ) = δ ( k k ). 5. As discussed in class and in Shankar, the classical Hamiltonian is de- fined by H = n summationdisplay i =1 p i ˙ x i − L , (1) where L is a function of the n coordinates x i and n velocities ˙

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Unformatted text preview: p i = ∂ L /∂ ˙ x i . Show that if L = T − V , and T = ∑ i ∑ j T ij ˙ x i ˙ x j , where the T ij ’s are constants, then H = T + V . You may assume with no loss of generality that T ij = T ji . 1 6. Shankar, problem 2.7.2 (i). 7. OPTIONAL; NOT TO BE TURNED IN: Suppose we have a particle in one dimension, whose Lagrangian is L = − mc 2 r 1 − ˙ x 2 /c 2 − V ( x ) , (2) where m and c are positive constants. (Actually, m is the rest mass and c is the speed of light.) Assume | ˙ x | < c . (a). Find the Lagrange equation of motion for this particle. (b). Find the canonical momentum p = ∂ L /∂ ˙ x . (c). Find the corresponding Hamiltonian H . (d). What is H in the limit | ˙ x | ≪ c ? 2...
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p827aps3 - p i = ∂ L/∂ ˙ x i Show that if L = T − V...

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