Question

Question 1 a. An engineer designing an electron microscope wants to ensure that the final instrument can resolve features separated by 10-9 m and therefore decides that the de Broglie wavelength of the electrons should be 50 times shorter, i.e. 2 x 10-11 m. NOTE: in answering this question, ignore relativistic effects and assume that classical mechanics applies.

i. What would be the speed of the electrons? [1 mark]

ii. What voltage must be used for accelerating the electrons to achieve this objective? [2 marks]

iii. Estimate the probability that these electrons can tunnel through a barrier of thickness 10-9 m, if the barrier is 5 eV higher than the electron’s kinetic energy? You may use the approximate expression for barrier tunnelling T ! e−2bL [4 marks]

iv. If there is a 10% uncertainty in the speed of the electrons, what is the minimum uncertainty in their position? [3 marks]

b. i. Why are the energy levels of a confined electron quantised (e.g. in an infinite potential well), while a free electron can take a continuous range of energies? [2 marks]

ii. An electron is confined to an infinite potential well of width 0.25 nm. What is the lowest energy (ground state) for the confined electron? [2 marks]

iii. Calculate the energy of the first and second states of the electron in the infinite potential well (i.e. n = 2,3,4) [2 marks]

iv. Calculate the wavelength of the photon emitted when an electron in the first excited state drops to the ground state. [1 mark]

v. Qualitatively describe the difference in the energy levels of the infinite potential well we have been investigating, and a finite potential well of depth 50 eV. Your answer should include a sketch of the probability density for an electron in the ground state for both the infinite and finite potential well.

Answer 1

Part B:

When an electron got stuck in a potential barrier, only those frequencies which match the boundary conditions of the barrier will sustain giving rise to a discreteness in the energy levels (It should be remembered always that constraining something gives rise to discreteness in the energy states). An infinite potential potential well is similar to the string tied at two ends which can resonate at only the harmonics of the fundamental frequency which arises due to the constraint that at the boundaries the displacement of the string should be zero. In the same way for an infinite potential well, the wavefunction of an electron should go to zero at the boundaries.

(ii) the energy levels of an electron inside a potential barrier of thickness of 0.25nm can be calculated using the formula which is derived from the schrodinger equation.

n = 1,2,3,4....

here L is the length of the barrier.

for lowest energy level n=1, by substituting the respective values the energy values can be easily calculated.

(iii) Use the same energy equation and calculate the energy levels at n=2,3,4.

(iv) The energy of the photon that is emitted should match with the energy gap of the two levels (here the electron jumps from the 1st excited state to the ground state)

Energy of the photon is given by

since = frequency of the photon.

(v) When the potential barrier becomes finite the electron has a exponential decaying tail which is evanescent tail which goes into the barrier and the decay rate is proportional of the depth of the barrier and this evanescent tail gives the probability for an electron to tunnel through the barrier.

Hope I have your question. please comment if there are any queries. Thank you.