Introduction to Quantum Mechanics · Problem 6.7

Symmetries and Conservation Laws — David J. Griffiths

Physics — Quantum Mechanics · translational_symmetry

Problem

Consider two particles of mass m1m_1 and m2m_2 (in one dimension) that interact via a potential that depends only on the distance between the particles V(∣x1−x2∣)V(|x_1 - x_2|), so that the Hamiltonian is H^=−ℏ22m1∂2∂x12−ℏ22m2∂2∂x22+V(∣x1−x2∣).\hat{H} = -\frac{\hbar^2}{2m_1} \frac{\partial^2}{\partial x_1^2} - \frac{\hbar^2}{2m_2} \frac{\partial^2}{\partial x_2^2} + V(|x_1 - x_2|). Acting on a two-particle wave function the translation operator would be T^(a)ψ(x1,x2)=ψ(x1−a,x2−a).\hat{T}(a) \psi(x_1, x_2) = \psi(x_1 - a, x_2 - a). (a) Show that the translation operator can be written T^(a)=e−iaℏP^,\hat{T}(a) = e^{-\frac{ia}{\hbar} \hat{P}}, where P^=p^1+p^2\hat{P} = \hat{p}_1 + \hat{p}_2 is the total momentum. (b) Show that the total momentum is conserved for this system.

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