An electron is trapped in a finite well. How “far” (in eV) is it from being free (that is, no longer trapped inside the well) if the penetration length of its wave function into the classically forbidden region is 1nm? The answer should be 0.038eV
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An electron is trapped in a finite well. How “far” (in eV) is it from being free (that is, no longer trapped inside the
well) if the penetration length of its wave function into the classically forbidden region is 1nm?
The answer should be 0.038eV
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- An electron with an energy of 8.0eV is incident on a potential barrier which is 9.2eV high and 0.25 nm wide. (a) What is the probability that the electron will pass through the barrier? (b) What is the probability that the electron will be reflected? (c) What is the wavelength of the electron before it encounters the barrier? (d) What is the wavelength of the transmitted electron?a question of quantum mechanics: Consider a particle in a two-dimensional potential as shown in the picture Suppose the particle is in the ground state. If we measure the position of the particle, what isthe probability of detecting it in region 0<=x,y<=L/2 ?The quantum mechanical tunneling process occurs if an electron is incident on a potential barrier of finite width a and finite height Uo. Calculate the transmission probability for this case assuming a barrier width a=1.4 nm and a barrier height Uo=2.6eV assuming that the energy of the electron is E=2.2eV. Give your result in % and round it off to two decimal places, i.e. the nearest hundredths. U(x) Uo 0 < E < Uo E 01 a
- An electron with an initial kinetic energy of 1.542 eV (in a region with 1.095 eV potential energy) is incident on a potential step (extending from x=0 to ∞) to V=2.381 eV. What is the transmission probability (in %)? FYI: If we had a travelling wave arriving at a similar potential DROP, then k1 (for x<0) would be real and the symmetry of R=(k1-k2)2/(k1+k2)2 implies reflection/transmission are the same as a potential RISE with the same energies but k1 and k2 swapped.Consider a particle moving in a one-dimensional box with walls at x = -L/2 and L/2. (a) Write the wavefunction and probability density for the state n=1. (b) If the particle has a potential barrier at x =0 to x = L/4 (where L = 10 angstroms) with a height of 10.0 eV, what would be the transmission probability of the electrons at the n = 1 state? (c) Compare the energy of the particle at the n= 1 state to the energy of the oscillator at its first excited state.An electron is trapped in an infinitely deep one-dimensional well of width 10 nm. Initially, the electron occupies the n = 4 state. Calculate the photon energy required to excite the electron in the ground state to the first excited state.
- = = An electron having total energy E 4.60 eV approaches a rectangular energy barrier with U■5.10 eV and L-950 pm as shown in the figure below. Classically, the electron cannot pass through the barrier because E < U. Quantum-mechanically, however, the probability of tunneling is not zero. Energy E U 0 i (a) Calculate this probability, which is the transmission coefficient. (Use 9.11 x 10-31 kg for the mass of an electron, 1.055 x 10-34] s for h, and note that there are 1.60 x 10-19 J per eV.) (b) To what value would the width L of the potential barrier have to be increased for the chance of an incident 4.60-eV electron tunneling through the barrier to be one in one million? nmConsider a particle in the n = 1 state in a one-dimensional box of length a and infinite potential at the walls where the normalized wave function is given by 2 nTX a y(x) = sin (a) Calculate the probability for finding the particle between 2 and a. (Hint: It might help if you draw a picture of the box and sketch the probability density.)A particle is in a three-dimensional cubical box that has side length L. For the state nX = 3, nY = 2, and nZ = 1, for what planes (in addition to the walls of the box) is the probability distribution function zero?
- The wavefunction for a quantum particle tunnelling through a potential barrier of thickness L has the form ψ(x) = Ae−Cx in the classically forbidden region where A is a constant and C is given by C^2 = 2m(U − E) /h_bar^2 . (a) Show that this wavefunction is a solution to Schrodinger’s Equation. (b) Why is the probability of tunneling through the barrier proportional to e ^−2CL?At time t = 0 the normalized wave function for a particle of mass m in the one-dimensional infinite well (see first image) is given by the function in the second image. Find ψ(x, t). What is the probability that a measurement of the energy at time t will yield the result ħ2π2/2mL2? Find <E> for the particle at time t. (Hint: <E> can be obtained by inspection, without an integral)