Consider a particle of mass m moving in one dimension under the influence of a constant...
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Consider a particle of mass m moving in one dimension under the influence of a constant gravitational acceleration g. Then the corresponding potential energy is given by V(x) = mgx and the Lagrangian of the mechanical system becomes L(x,x) = -mi² - mgx Like the free particle and the harmonic oscillator cases, this model can be solved exactly. (a) Solve for the classical trajectory ret(t) which obeys the Euler-Lagrange equa- tion: d ƏL ƏL ????х dt di (b) Calculate the classical action S[ra(t)], where the classical action is defined S[z(t)] = [*¹ dtL (x, i) = 0 for a path satisfying the boundary condition (ti) = T₁, Te(tf) = xf. (c) Now imagine performing the path integral for the transition amplitude (prop- agator) by expansing about the classic trajectory r(t) = x(t) + 8x(t) Demonstrate that the transition amplitude (x, ts|xi, ti} = [ Dx(t)eis[=(1)] = (0, ts|0, †₁)0€*S[=ca (6)] (i) where (0, t0,t₁)o is the transition amplitude for the free particle, which has been solved in class: (0, tf 0, ti)o= = m (2πi(tf-ti) 1/2 Consider a particle of mass m moving in one dimension under the influence of a constant gravitational acceleration g. Then the corresponding potential energy is given by V(x) = mgx and the Lagrangian of the mechanical system becomes L(x,x) = -mi² - mgx Like the free particle and the harmonic oscillator cases, this model can be solved exactly. (a) Solve for the classical trajectory ret(t) which obeys the Euler-Lagrange equa- tion: d ƏL ƏL ????х dt di (b) Calculate the classical action S[ra(t)], where the classical action is defined S[z(t)] = [*¹ dtL (x, i) = 0 for a path satisfying the boundary condition (ti) = T₁, Te(tf) = xf. (c) Now imagine performing the path integral for the transition amplitude (prop- agator) by expansing about the classic trajectory r(t) = x(t) + 8x(t) Demonstrate that the transition amplitude (x, ts|xi, ti} = [ Dx(t)eis[=(1)] = (0, ts|0, †₁)0€*S[=ca (6)] (i) where (0, t0,t₁)o is the transition amplitude for the free particle, which has been solved in class: (0, tf 0, ti)o= = m (2πi(tf-ti) 1/2
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Intermediate Accounting principles and analysis
ISBN: 978-0471737933
2nd Edition
Authors: Terry d. Warfield, jerry j. weygandt, Donald e. kieso
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