Answered: Consider the normalized wave function,… | bartleby

Related questions

Question

Consider the normalized wave
function,
$(x)= c₁4₁(x) + 4₂(x) + √243(x),
1
1
2
૩(૩),
1
√8
where u̟₁(x), ₂(x), and u̟²(x) are the
normalized eigenfunctions of H.
a) Solve for C₁.
b) Calculate (E) in terms of E₁, E2, and E3.
c) If we measure the energy of a single system,
what is the probability that it will be E₁?

Expert Solution

Step by step

Solved in 3 steps with 3 images

SEE SOLUTION Check out a sample Q&A here

Blurred answer

Similar questions

You are given a free particle (no potential) Hamiltonian Ĥ dependent wave-functions = -it 2h7² m sin(2x) e = V₁(x, t) V₂(x, t) 2 sin(x)e -ithm + sin(2x)e¯ What would be results of kinetic energy measurements for these two wave-functions? Give only possible outcomes, for example, it is possible to get the following values 5, 6, and 7. No need to provide corresponding probabilities. ħ² d² 2m dx2 and two time- -it 2hr 2 m
da=do= 4) For the following d hydrogenic wavefunctions, find the magnitude and the z component of the orbital angular momentum. You can give your results in terms of h. اور اہل = 1 √2 16t i√2 i√2 Final (d₁2+ d_2) = (165) R₁2(r) (x²- y²)/r² 151/2 1/2 R₁2(r) (3cos²0-1)=(16) R2(1)(32²-7²)/² 1/2 (d+2-d-2)= (15) Rm2(7)xy/² =(d. 1+d-1)= (d_1-d-1)=- 1/2 1/2 (15) R₁2 (r)yz/r² 10 June 2011 1/2 (45) * Ru2(1) 2x/r²
40. The first excited state of the harmonic oscillator has a wave function of the form y(x) = Axe-ax². (a) Follow the
please answer c) only 2. a) A spinless particle, mass m, is confined to a two-dimensional box of length L. The stationary Schrödinger equation is - +a) v(x, y) = Ev(x, y), for 0 < r, y < L. The bound- ary conditions on ý are that it vanishes at the edges of the box. Verify that solutions are given by 2 v(1, y) sin L where n., ny = 1,2..., and find the corresponding energy. Let L and m be such that h'n?/(2mL²) = 1 eV. How many states of the system have energies between 9 eV and 24 eV? b) We now consider a macroscopic box (L of order cm) so that h'n?/(2mL?) ~ 10-20 eV. If we define the wave vector k as ("", ""), show that the density of states g(k), defined such that the number of states with |k| between k and k +dk is given by g(k)dk, is Ak 9(k) = 27 c) Use the expression for g(k) to show that at room temperature the partition function for the translational energy of a particle in a macroscopic 2-dimensional box is Z1 = Aoq, where 2/3 oq = ng = mk„T/2nh?. Hence show that the average…
Consider a particle in a 2-D box having Lx = 10 nm and Ly = 10 nm. a) Make a surface plot of all the wave functions for the first and second energy levels. b) What is the degeneracy of the second energy level? Compare and contrast the wave functions of the second energy level. c) How does the number of nodes in the x-coordinate change as n increases? How does the number of nodes in the y-coordinate change as n, increases? d) Explain whether or not those same states would be degenerate if Lx = 10 nm and Ly = 15 nm.
A qubit is in state |) = o|0) +₁|1) at time t = 0. It then evolves according to the Schrödinger equation with the Hamiltonian Ĥ defined by its action on the basis vectors: Ĥ0) = 0|0) and Ĥ|1) = E|1), where E is a constant with units of energy. a) Solve for the state of the qubit at time t. b) Find the probability to observe the qubit in state 0 at time t. Explain the result by referring to the way that the time-evolution transforms the Bloch sphere.
Consider an electron trapped in a 20 Å long box whose wavefunction is given by the following linear combination of the particle's n = 2 and n = 3 states: ¥(x,t) =, 2nx - sin ´37x - sin 4 where E, 2ma² a a. Determine if this wavefunction is properly normalized. If not, determine an appropriate value for a normalization constant. b. Show that this is not an eigenfunction to the PitB problem. What are the possible results that could be returned when the energy is measured and what are the probabilities of measuring each of these results?
Consider the superposition wave function (x) = c sin (πx/a) + d sin (2πx/a). a. Is (x) an acceptable wave function for the particle in the box? b. Is(x) an eigenfunction of the total energy operator Ĥ? c. Is(x) normalized?
The wave function of a particle at time t= 0 is given by w(0) = (4,) +|u2})), where |u,) and u,) the normalized eigenstates with eigenvalues E and E, are respectively, (E, > E, ). The shortest time after which y(t) will become orthogonal to |w(0)) is - ħn (а) 2(E, – E,) (b) E, - E, (c) E, - E, (d) E, - E,