4. Suppose the speed of a projectile of mass 1.0 g is known to within 1.0 µms-1. Calculate the minimum uncertainty in its position. Repeat the computation for an electron and comment on the results.
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- 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?Help
- Q3. Given: mass of electron = 9.1x10-31/C; e = 1.60x10-19C A. Prove that if an electron is accelerated from rest by a potential of 54v, de Broglie wavelength is 5.284x10-13. B. If λ = h/mv, use de Broglie wavelength in a 10-g bullet moving at 400 m/s1. An electron initially moves in first excited state, if it absorb energy with frequency 5x10te Hz. Find the momentum of this electron before and after interaction with photon energy.1. A free electron has the wavefunction p(x, t) = sin(kx – wt) and wave number k = 50nm-1 1.1 Determine the electron's de Broglie wavelength. 1.2 Determine the electron's momentum. 1.3 Determine the electron's kinetic energy. 2. An electron is trapped in an infinite potential well of width 1 cm. For what value of n will electron have an energy of 2 eV? 3. Give the Born interpretation of the wavefunction.
- 4. An electron has a momentum py =1.30 x 10 kg.m/s. What is the minimum uncertainty in its position that will keep the relative uncertainty in its momentum (Apy/p) below 2.3%? (35.3 nm6. An electron in hydrogen atom is in initial state p(r, 0) = A(2µ100 + ¡Þ210 + 4Þ21–1 – 2i4211) where ynim are the eigenfunctions of the hydrogen atom a. Determine the constant A b. What is the probability of finding the electron in the first excited state? c. Write the state p(r, t) at time t, using energy eigenvalues as En = d. Find the expectation value of L in the state (r,t e. Find the expectation values of Lx and Ly in the state (r, t f. If measurement of L, led to the value -ħ what will be results of measurement of energy and the square of total orbital momentum immediately afterwards and what are their probabilities? hw n24. An H-atom is in is state. a. What is the probability that the electron is found in sphere given by r ≤ ao b. What is (r) ? c. calculate ().
- 4. A linear oscillator of frequency stays in an environment of absolute temperature T. What is the probability that the oscillator is in its first excited state (with energy)?4. Show that if f= x/y, then the uncertainty in f can be written as: slee + σ y1. Consider a quantum system in state |) = ¢1) +l¢2)+l¢3) expressed in terms of the orthonormal eigenvectors {|Øn}} of operator, Q, where its spectrum of allowed measured values is given by Qn = n². If many different experiments were performed, what is the expected average outcome of quantity, Q?