The vector potential is expanded in momentum modes as A= h (paxpep-/h+ea{Pe-ip-/h 2eoL3w (1) h Ai=-i e{papei/h-epae-ip/h. (2 V 2egL3wp D.A a Show that the Hamiltonian is [20] =hwpapaxp p,x (3) where the scalar potential φ is zero
Added by Matthew E.
Close
Step 1
First, let's calculate the curl of the vector potential A using equation (1): ∇ x A = h ∇ x (paxpe^(-ipx/h)) + e∇ x (aPe^(-ipx/h)) Using the identity ∇ x (fA) = (∇f) x A + f(∇ x A), where f is a scalar function and A is a vector, we can rewrite the above equation Show more…
Show all steps
Your feedback will help us improve your experience
Mahipal Kumawat and 73 other Physics 103 educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
(a) Write down (i) the Hamiltonian operator, Ĥ (where Ĥ = T̂ + V̂), for a general potential V(x). (ii) the operator for momentum, p̂. (b) Evaluate the commutator [Ĥ, p̂] of these operators and comment on the significance of the result. (c) How would your result for (b) be affected if V(x) = 0, i.e. for a free particle? (d) It can be shown that the time evolution of the expectation value ⟨O⟩ of an observable represented by some general operator Ô is given by d⟨O⟩/dt = ⟨i/ħ [Ĥ, Ô]⟩ where Ĥ is the Hamiltonian operator you wrote down in part (a) and [Ĥ, Ô] is the commutator of the operators Ĥ and Ô. Use this result and the commutator you obtained in part (b) to show that d⟨p⟩/dt = -⟨dV(x)/dx⟩, which is special case of Ehrenfest's Theorem. What is the significance of this result?
Adi S.
The Plane Harmonic oscillator has the Hamiltonian H = 1/(2m)(p_x^2 + p_y^2) + 1/2 mΔ^2(q_x^2 + q_y^2) a) Find the energy levels and their degeneracy; b) express the Hamiltonian in terms of the operators η_+ = 1/√2(a_x + ia_y) η_- = 1/√2(a_x - ia_y) where a_x = √(mω/2ℏ)q_x + i√(1/2mωℏ)p_x a_y = √(mω/2ℏ)q_y + i√(1/2mωℏ)p_y and their Hermitian adjoint. c) write the angular momentum operator for this system; what can we say about the angular momentum at a fixed energy eigenvalue?
Madhur L.
The Hamiltonian operator for H2+ in the Born-Oppenheimer approximation is: (A) H = +1/2∇² - 1/r_A - 1/r_B + 1/R (B) H = -1/2∇² + 1/r_A + 1/r_B - 1/R (C) H = -1/2∇² - 1/r_A - 1/r_B + 1/R (D) H = +1/2∇² - 1/r_A + 1/r_B - 1/R
Jiva Y.
Recommended Textbooks
University Physics with Modern Physics
Physics: Principles with Applications
Fundamentals of Physics
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD