3.12.2 We perform an experiment in which we prepare a particle in a given quantum mechanical...
A quantum particle is prepared in the state |ψ> = 0.8 |E1> + 0.6 i |E2>. where |E1> and |E2> are energy basis states. A measurement is made of the particle's energy. What is the probability of measuring the value E2?
3 Angular Momentum and Spherical Harmonics For a quantum mechanical system that is able to rotate in 3D, one can always define a set of angular momentum operators J. Jy, J., often collectively written as a vector J. They must satisfy the commutation relations (, ] = ihſ, , Îu] = ihſ, J., ſu] = ihỈy. (1) In a more condensed notation, we may write [1,1]] = Žiheikh, i, j= 1,2,3 k=1 Here we've used the Levi-Civita symbol, defined as...
11. A quantum particle in an infinitely deep square well be QIC wave function given by 9x sin for 0 SXS Land zero otherwise, (a) Determine the expec. tation value of x. (b) Determine the probability of finding the particle near L by calculating the probability that the particle lies in the range 0.490L < x 0.510L. (c) What If? Determine the probability of finding the particle near L by calculating the probability that the particle lies in the range...
Quantum Mechanics question about an infinite square well. A particle in an infinite square well potential has an initial state vector 14() = E1) - %|E2) where E) is the n'th eigenfunctions of the Hamiltonian operator. (a) Find the time evolution of the state vector. (b) Find the expectation value of the position as a function of time.
A particle moves in an infnite potential well described by V(r) o, l> a/2. are of the forn vn (z)-A" cos (k,,e), or Un(r) B," sin (knz), depending on the value of n. For n 3, (r)-(V2/a) cos (3Tr/a) for lrl S a/2 and var t are the expectation values of r and a2 in the n 3 state. ) What are the expectation values of p and p2 in the n-3 state. To calculate the expectation value for momentum,...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
A particle moves in an infinite potential well described by The eigenfunctions are of the form (r) = A For n = 3. e3(r) = (v/2/n) cos(3mr/n) for lrl cos (knr), or er (r) = Dn sin (k, r), depending E0 for o/2 and t's(r)- (a) What are the expectation values of r and 2 in the n 3 state. (b) What are the expectation values of p and p2 in the n 3 state. To calculate the expectation value...
plz help. thnx The state of a quantum mechanical particle, constrained to move on a circle of radius R in the x-y plane, is given by 4. where ф is the angle that the position vector makes with the x-axis a) Find a value of N which makes the above state normalized b) If Lz is measured, what are the possible outcomes and their corresponding probabilities?
Question #2: 6 pts] Find the eigenvalues and the normalized eigenvectors of the matrix 21 2 -1 2 Question #3: 10 pts] The electron in a hydrogen atom is a linear combination of eigenstates. Let us assume a limited linear combination to provide some sample calculations $(r, θ, φ) 2 ,1,0,0 + '2,1,0 (a) Normalize the above equation. (b) What are the possible results of individual measurements of energy, angular momentum, and the z-component of angular momentum? (c) What are...
[12 6. Consid er a particle of mass m moving in an infinitely deep square well potential of width a, whose wave function at time t 0 is where on Ce) is the normaized wave function of the n-th eigenstate of the Hamitonian of that particle The corresponding eigen-energy of the n-th state is 2ma?n 1,2,3,... (e) Find the average energy of the system (ie. the expectation value () (b) Write down the wave function p(z,t) at a later time...