The Hamiltonian of a perturbed 2-dimensional oscillator is given by . Prove that can be diagonalized by the unitary transformation , where is a parameter. Find also the values of such that the transformation is well-defined.
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4. (30 points) Harmonic oscillator with perturbation Recall the Hamiltonian of an harmonic oscillator in 1D: p21 ÃO = + mwf?, where m is the mass of the particle and w is the angular frequency. Now, let us perturb the oscillator with a quadratic potential. The perturbation is given by Î' = zgmw?h?, where g is a dimensionless constant and g <1. (a) Write down the eigen-energies of the unperturbed Hamiltonian. (b) In Lecture 3, we introduced the lowering (or...
1. Problems on unitary operators. For a function f(r) that can be expanded in a Taylor series, show that Here a is a constant, and pis the momentum operator. The exponential of an operator is defined as ea_ ??? i,O" Verify that the unitary operator elo/h can be constructed as follows (Hint: Notice that f(x +a) (al) and eohf())) e Prove that Here is the position operator. (Hint: You may work in the momentum space, in which p = p...
Problem 7.49 Problem 7.49 A hydrogen atom is placed in a uniform magnetic field Bo Bo (the Hamiltonian can be written as in Equation 4.230). Use the Feynman-Hellman theorem (Problem 7.38) to show that a En (7.114) where the electron's magnetic dipole moment10 (orbital plus spin) is Yo l-mechanical + γ S . μ The mechanical angular momentum is defined in Equation 4.231 a volume V and at 0 K (when they're all in the ground state) is41 Note: From...
Question A2: Coherent states of the harmonic oscillator Consider a one-dimensional harmonic oscillator with the Hamiltonian 12 12 m2 H = -2m d. 2+ 2 Here m and w are the mass and frequency, respectively. Consider a time-dependent wave function of the form <(x,t) = C'exp (-a(x – 9(t)+ ik(t)z +io(t)), where a and C are positive constants, and g(t), k(t), and o(t) are real functions of time t. 1. Express C in terms of a. [2 marks] 2. By...
6. For each of the following functions, either prove that it is a linear transformation, or show that it is not. If the given transformation is indeed linear, find the matrix of the transformation (a) TR2R2 where T "folds" the half of the plane below the diagonal line z2 to the top half. In other words: T1 3x1- 2 (b) T:R3-R3 where T3x2 - 3x3- 1
7 Harmonic oscillator in "energy space" Consider the harmonic oscillator in "energy space", i.e., in terms of the basis of eigenvectors n) of the harmonic oscillator Hamiltonian, with Hn) -hwn1/2)]n). We computed these in terms of wavefunctions in position space, ie. pn(x)-(zln), but we can also work purely in terms of the abstract energy eigenvectors in Dirac notation. PS9.pdf 1. You computed the matrix elements 〈nleln) on an earlier problem set. Now find (nn) for general n,n' 2. Find the...
Consider a particle of mass m under the action of the one-dimensional harmonic oscillator potential. The Hamiltonian is given by Knowing that the ground state of the particle at a certain instant is described by the wave function mw 1/4 _mw2 Th / calculate (for the ground state): a) The mean value of the position <x> (2 marks) b) The mean value of the position squared < x2 > (2 marks) c) the mean value of the momentum <p> (2...
3. (10 points) The Hamiltonian for a particle moving in a vertical uniform gravitational field g is 2 Hmgq where q is the altitude above the ground. We want to find any canonical transformation from old variables (q,p) to new variables (Q,P) which provides a cyclic coordinate. To do this, define new variables as where a, b are constants. a) Determine any combination of constants a and b, which provides a canonical transformation b) Find the type 1 generating function,...
11 Consider the one-dimensional oscillator defined on page 6. At time t function (r, t is given by 0, its wave (a, 0) N{(2+3i)óo(a) - V/5¢2(x) + (2 - i/3)()} (a) Choose N such that Į is normalised to 1. [2] (b) What are the allowed energies, and with what probabilities? (c) What is the wave function at time t? [] What is the probability for even parity to be measured? Briefly explain (d) [I] explicit expression for a_ V...
Consider the dimensionless harmonic oscillator Hamiltonian, (where m = h̄ = 1). Consider the orthogonal wave functions and , which are eigenfunctions of H with eigenvalues 1/2 and 5/2, respectively. with p=_ïda 2 2 We were unable to transcribe this imageY;(r) = (1-2x2)e-r2/2 (a) Let фо(x-AgVo(x) and φ2(x) = A2V2(x) and suppose that φ。(x) and φ2(x) are normalized. Find the constants Ao and A2. (b) Suppose that, at timet0, the state of the oscillator is given by Find the constant...