If the wave function has the following equation y = 4cosA + 4isinA a. Determine the conjugate b. |y*y|
Q: 1. A particle of m moves in the attractive central potential: V(r) = ax6, where a is a constant and…
A: Ans 1: (a) A=(π2b)1/4. (b) E(b)=2mbℏ2+64b315α. (c) bmin=(32ℏ245αm)1/4. (d)…
Q: A particle of mass m moves freely in a one-dimensional box of length 3a. In the same diagram, sketch…
A:
Q: -x² wave function y(x) = € 3², (−∞0 ≤ x ≤ +∞). If the wave function is not normalized, please…
A:
Q: 3. Consider a particle in an infinite square well potential trapped between 0 < x < a. What is the…
A: The energy eigen values of a particle confined in an infinite well with width a is given by…
Q: With a value of 1 = 1 and the angular momentum component Ly is known below. If the state operator Ly…
A: Given: For l= 1 Angular momentum componenet Ly in the matrix formation…
Q: 2. Find the best bound on Es for the one-dimensional harmonic oscillator using the trial wave…
A:
Q: a 2 otherwise
A: The calculation is shown below,
Q: Q1: Find the first excited state of harmonic oscillator using the equation: ₂(x) = A (a*)(x). with…
A:
Q: If ø(x) is an arbitrary well-behaved function, can one claim [î, §]= iħ1 ? (YES or NO). Give reasons…
A:
Q: 2. Consider a density operator p. Show that tr (p²) < 1 with tr (p²) = 1 if and only if p is a pure…
A: Introduction: Consider an ensemble of given objects in the states. If all the objects are in the…
Q: A particle is described by the following normalized superposition wavefunction: Y(x)= =√(si…
A:
Q: Q4. Proof that the probability current for a wavefunction of the form (a) j = R(x)|² h ³s(x) m əx…
A: Since we answer up to one question, we will answer the first question only. Please resubmit and…
Q: The figure below shows the square of the wavefunction of a particle confined in a one-dimensional…
A:
Q: For a one-dimensional box, we assume that the particle is confined between rigid, unyielding walls…
A:
Q: 1. A particle of m moves in the attractive central potential: V(r) = ax6, where a is a constant and…
A: The objective of the question is to compute the normalization constant A, calculate the ground state…
Q: A particle in a 3-dimensional quadratic box with box length L has an energy given by (n+n+n2). The…
A: Degeneracy: Degeneracy can be defined as the number of states having the same energy.Formula for the…
Q: 4. Find the points of maximum and minimum probability density for the nth state of a particle in a…
A: For a 1-D box The wave function is, ψnx=2L sinnπxLProbability density, ρ=ψ*nψn =2L sinnπxL2L…
Q: 2. A wave function is a linear combination of 1s, 2s, and 3s orbitals: y(r) = N(0.25 w,; + 0.50w2;…
A: Given: A wave function is a linear combination of 1s, 2s, and 3sψ(r) = N0.25ψ1s + 0.50ψ2s + 0.30 ψ3s…
Q: 1. Given the following probability density function: p(x) = Ae-A(x-a)². %3D 2. A particle of mass,…
A: Expectation Value: The expectation value in quantum mechanics measures the expected value of any…
Q: 2. A particle is confined to the x-axis between x = 0 and x = 3a. The wave function of the particle…
A: The detailed solution is following.
8. If the wave function has the following equation y = 4cosA + 4isinA
a. Determine the conjugate
b. |y*y|
Step by step
Solved in 2 steps with 1 images
- 7. Consider a particle in an infinite square well centered at x = 0 in one of its stationary states. For this problem, you may look up any integrals. Some useful ones are given in Harris. a) Compute (x) and (pr) for arbitrary n. Do this by direct computation but then describe how you could have found these results using symmetry (the symmetry can either be symmetry in the physical system, such as the shape of the wave function, or symmetry related to the expectation value integral, such as the shape of the integrand). b) Using your answer to part a), show that the uncertainty in the momentum is Apx nh for arbitrary n. Do this two ways: (i) first by using your answer to part a) and directly computating (p2) (via an integral) and (ii) by using your answer to part a) and relating (p2) to the kinetic energy operator. c) Show that the uncertainty principle holds for the ground state. 2L -3. A system of N harmonic oscillators of frequency w are prepared in identical initial states of wavefunction Þ(x,0). It is found that the measurement of the energy of the system at t = 0 gives 0.5hw with probability 0.25, 1.5hw with probability 0.5 and 2.5hw with probability 0.25. a. Write a possible function p(x, 0). b. Write the corresponding (x, t). c. What is the expectation value of the Hamiltonian in the state p(x,t) ? d. Calculate the expectation value of position at time tQUESTION 6 Consider a 1-dimensional particle-in-a-box system. How long is the box in radians if the wave function is Y =sin(8x) ? 4 4л none are correct T/2 O O O
- 3. Plane waves and wave packets. In class, we solved the Schrodinger equation for a "free particle" (e.g. when U(x,t) = 0). The correct[solution is (x, t) = Ae(px-Et)/ħ This represents a "plane wave" that exists for all x. However, there is a strange problem with this: if you try to normalize the wave function (determine A by integrating * for all x), you will find an inconsistency (A has to be set equal to 0?). This is because the plane wave stretches to infinity. In order to actually represent a free particle, this solution needs to be handled carefully. Explain in words (and/or diagrams) how we can construct a "wave packet" from the plane wave solution. (Hint 1: consider a superposition of plane waves for a limited range of momentum/energy. Hint 2: have a look at the brief discussion in the middle of pg. 278 and especially pg. 308-309 of the text.)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?