Exercise 12. Use Heisenberg's Uncertainty Principle to find an order of magnitude estimate for the minimum kinetic energy of an electron that is confined to a one-dimensional region of extent 1 x 10-7 m.
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![Exercise 12.
Use Heisenberg's Uncertainty Principle to find an order of magnitude estimate
for the minimum kinetic energy of an electron that is confined to a one-dimensional region of
extent 1 x 10-7
m.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F5b5c27d4-e1be-40a6-96b1-54faeab744dc%2Fdf9b09df-b896-4aef-9caa-6e7448156176%2F6dprc6pi_processed.jpeg&w=3840&q=75)
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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.Calculate the de Broglie wavelength (in fm) of a 6.5 MeV a particle emitted from an atomic nucleus whose diameter is approximately 1.8 x fm Calculate its minimum kinetic energy (in keV) according to the uncertainty principle. kev 10-14 m.3. For two electrons in a 1-dimensional well of length L (a) Write the properly normalized, asymmetric wave function for n = 1, n'=2. (b) Calculate the probability that both electrons will be found at x <4.
- 2. A proton with kinetic energy 1.00 eV is incident on a square potential barrier with height 10.00 eV. If the proton is to have the same transmission probability as an electron of the same energy, what must the width of the barrier be relative to the barrier width encountered by an electron?Calculate the de Broglie wavelength for: a) An electron (9.11 x 10-31) moving at 1 x 106 m/s b) A neutron (1.67 x 10-27 kg) moving at 2.6 x 104 m/s Both particles approach a potential barrier (U = 5 eV) 0.26 nm wide. Calculate the transmittance coefficient for each particle. %3D7. 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. 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. 5. Suppose the wavefunction, b(x), is an Eigenfunction of the operator Q with an Eigenvalue Q. Show that the uncertainty in the measurement of Q is given by AQ = V(Q?) – (Q)². Assume (x) is normalized.1. 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?