1. Find the physical dimensions of wave function ψ(r) of a particle moving in three dimensional spaces.
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1. Find the physical dimensions of wave function ψ(r) of a particle moving in three dimensional
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- 3. A harmonic oscillator of mass m and angular frequency w is in the initial state of wavefunction p(x,0) = Ai4o(x) + 2Ai¢2(x) Obtain the constant A b. Write the function (x, t) c. Calculate the uncertainties Ax and Ap in the state of wavefunction p(x,t) and show that the Heisenberg uncertainty principle is satisfied a.3. A particle is confined to x-axis the between x = 0 and x = L. The wave function of the particle is = Ae¹(x) with A € C. a. Determine A. b. G. Determine (p). Determine (E).3. For free particles in two dimensions, what is density of states (DOS) in low speed limit (=p²/2m), and in high speed limit (=pc)?
- 1. Louis de Broglie postulated all matter has both a particle and wave nature. The wavelength of any matter wave is given by λ = h/p, where p is the linear momentum p = mv and both p and v are vectors. a) World class marathon runners can run at an average pace of about 3.00 min/km. What is the wavelength, momentum and kinetic energy of a 55 kg runner racing at this average speed? b) What is the uncertainty in the runner's velocity if the position is to be determined within one wavelength?3. Particle in a 2D Box. A quantum mechanical particle is confined in side a square 2D box, with side length L. Inside the box V=0 and outside the box V=infinity. Let the wave function to be (x,y). (a) write down the Schrodinger equation of (x,y). (b) Use the separation of variable method solve (x,y) (let the quantum numbers to be nx and ny.) (c) What is the energy for the state (nx, ny)? (d) What is the probability density p(x,y) for the state nx=3 and ny=3? Sketch this p(x,y) in a square.7. One electron is trapped in a one-dimensional square well potential with infinitely high sides. a. If you have a probe that has a width for electron detection Ax = 0.00350L in the x direction, for the first excited state ( n =2), what is the probability that the electron is found in the probe when it is centered at x = L/4, (hint: you can use an approximation for this - you do not need to do an integral)? b. What is the average number of electrons that you would detect using the probe described in part "b." centered at x = L/4, ifthe electron is in the first excited state (n = 2) for each experiment and you repeat the experiment N, =100,000 times?
- 4. Use the variational principle to estimate the ground state energy of a particle in the potential (∞0 x < 0 U(x) = \cx x≥0 Take xe-bx as a trial function.A particle in a box, with the standing matter wave shown, has an energy of 8.0 eV. What is the lowest energy that this particle can have?A. 1.0 eV B. 2.0 eV C. 4.0 eV D. 8.0 eV4. Normalize the following wavefunctions 4 55 (a) v(x) = sin (#2); =sin(); for a particle in a 1D box of length L. (b) (2) = xe-z|2 (c) (x) = e(x²/a²)+(ikz) 5. In a region of space, a particle with mass m and with zero energy has a time- independent wave-function (x) = Ae-2/12, where A and L are constants. Use your knowledge of the Schrödinger equation to determine the potential energy V(x) of the particle. Plot the potential function? What is the minimum potential energy for the particle, if it is an electron and L = 1 fm? Is this potential repulsive or attractive?