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12. Write down the TISE in plane polar coordinates for a two-dimensional isotropic harmonic oscillator V(x,y) = zmo (x² +y°). where w is the angular frequency of the oscillator. Solve it and find the energy levels and the corresponding eigenfunctions. Discuss the degeneracy of the energy states. 12 (o) Colo to the nohlo digtono of the leotron from the uoloug in tho

12. Write down the TISE in plane polar coordinates for a two-dimensional isotropic harmonic oscillator V(x,y) = zmo (x² +y°). where w is the angular frequency of the oscillator. Solve it and find the energy levels and the corresponding eigenfunctions. Discuss the degeneracy of the energy states. 12 (o) Colo to the nohlo digtono of the leotron from the uoloug in tho

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Question 12: Quantum Mechanics
12. Write down the TISE in plane polar coordinates for a two-dimensional isotropic harmonic oscillator V(x,y) = zmo (x² + y°). where w is the angular frequency of the oscillator. Solve it and find the energy levels and the corresponding eigenfunctions. Discuss the degeneracy of the energy states. 13. (a) Calculate the most probable distance of the electron from the nucleus in the ground state of hydrogen. (b) Find the average distance of the electron from the nucleus in the ground state of hydrogen and compare it with the result of (a). Quantum Mechanics in Three Spatial Dimensions 237 14. Calculate (r²) in the ground state of the hydrogen atom. Using this and the result of the Problem 13, calculate the uncertainty in the measurement of the distance of the electron from the nucleus in the ground state of hydrogen. 15. The normalized ground state wave function for the electron in the hydrogen atom is 1 y(r,0,0) = παι ,-r/ao. where r is the radial coordinate of the electron and ao is the Bohr radius. (a) Sketch the wave function as a function of r. (b) Show that the probability of finding the electron between r and r+ dr is given by 4 -2r/ao dr. (c) Show that the wave function as given is normalized. (e) Find the probability of locating the electron between ao/2<r< 3ao/2. 16. Calculate (x) and (x²) in the ground state of the hydrogen atom. 17. The radial part of the wave function for the hydrogen atom in the 2p state is given by y(r, 0,ø) = Are¯r/2a0. where A is a constant and ao is the Bohr radius. Using this expression, calculate the average value of r for an electron in this state. 18. An electron in a hydrogen atom is in the energy eigenstate V2,1,–1(r, 0, ¢) = Nre "/240Y,-' (0,4). (a) Find the normalization constant N. (b) What is the probability per unit volume of finding the electron at r = 2ao, 0 = 45° and o = 60°?
Transcribed Image Text:12. Write down the TISE in plane polar coordinates for a two-dimensional isotropic harmonic oscillator V(x,y) = zmo (x² + y°). where w is the angular frequency of the oscillator. Solve it and find the energy levels and the corresponding eigenfunctions. Discuss the degeneracy of the energy states. 13. (a) Calculate the most probable distance of the electron from the nucleus in the ground state of hydrogen. (b) Find the average distance of the electron from the nucleus in the ground state of hydrogen and compare it with the result of (a). Quantum Mechanics in Three Spatial Dimensions 237 14. Calculate (r²) in the ground state of the hydrogen atom. Using this and the result of the Problem 13, calculate the uncertainty in the measurement of the distance of the electron from the nucleus in the ground state of hydrogen. 15. The normalized ground state wave function for the electron in the hydrogen atom is 1 y(r,0,0) = παι ,-r/ao. where r is the radial coordinate of the electron and ao is the Bohr radius. (a) Sketch the wave function as a function of r. (b) Show that the probability of finding the electron between r and r+ dr is given by 4 -2r/ao dr. (c) Show that the wave function as given is normalized. (e) Find the probability of locating the electron between ao/2<r< 3ao/2. 16. Calculate (x) and (x²) in the ground state of the hydrogen atom. 17. The radial part of the wave function for the hydrogen atom in the 2p state is given by y(r, 0,ø) = Are¯r/2a0. where A is a constant and ao is the Bohr radius. Using this expression, calculate the average value of r for an electron in this state. 18. An electron in a hydrogen atom is in the energy eigenstate V2,1,–1(r, 0, ¢) = Nre "/240Y,-' (0,4). (a) Find the normalization constant N. (b) What is the probability per unit volume of finding the electron at r = 2ao, 0 = 45° and o = 60°?
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