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Use the variational approch with the test function.
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¥(r,0,) = e-6 to determine the ground state energy of the hydrogen atom. In the equation,...
The wave function for a hydrogen atom in the ground state is given by
( 25 marks) The wave function for a hydrogen atom in the ground state is given by \(\psi(r)=A e^{-r / a_{s}}\), where \(A\) is a constant and \(a_{B}\) is the Bohr radius. (a) Find the constant \(A\). (b) Determine the expectation value of the potential energy for the ground state of hydrogen.
1) (60 points) The ground state of the hydrogen atom: In three dimensions, the radial part...
1) (60 points) The ground state of the hydrogen atom: In three dimensions, the radial part of the Schrodinger equation appropriate for the ground state of the hydrogen atom is given by: ke2 -ħ2 d2 (rR) = E(rR) 2me dr2 where R(r) is a function of r. Here, since we have no angular momentum in the ground state the angular-momentum quantum number /=0. (a) Show that the function R(r) = Ae-Br satisfies the radial Schrodinger equation, and determine the values...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in the spherical coordinate representation is where ao-top- 0.5298 10-10rn is the Bohr radius, and μ is the electon-proton reduced mass. Here, the square of the angular momentum operator L2 in the spherical coordinate representation is given by: 2 (2.2) sin θー sin θ 00 The form of the Schrödinger equation means that all energy eigenstates separate into radial and angular motion, and we can write...
e) A hydrogen atom is in its ground state (n = 1). Using the Bohr theory...
e) A hydrogen atom is in its ground state (n = 1). Using the Bohr theory of the atom, calculate (e) the energy gained by moving to a state where n = 5. g) A hydrogen atom is in its ground state (n = 1). Using the Bohr theory of the atom, calculate (g) the wavelength, λ, of the EM waved adsorbed in the process of moving the electron to a state where n = 5. Hint: There are two...
In the ground state of the Hydrogen atom the energy of the electron is E state...
In the ground state of the Hydrogen atom the energy of the electron is E state of the He ion? -13.61 eV. What is the energy of the electron in the ground Submit Answer Tries 0/20 What is the energy of the electron in the ground state of the u** ion? Submit Arower Tries 0/20 The electron in the Helon is excited to the n-2 principal state. What is the energy of the electron now? Submit Awer Tries 0/20 What...
45. Use a Gaussian trial function o(r) = e" to compute the ground state energy of...
45. Use a Gaussian trial function o(r) = e" to compute the ground state energy of the hydrogen atom. What is the "best" value of the variational parameter a? How far off is this energy from the true value?
For the hydrogen atom, its energy at ground state is 13.6 eV, at first excited state...
For the hydrogen atom, its energy at ground state is 13.6 eV, at first excited state is 3.4 eV at second excited state is 1.5 eV and at the third excited state is 0.85 eV. i) Give the energy value for the first two states in Joule (J). [1eV =1.6 x 10-19 J] (2 marks) ii) With the aid of schematic diagram, determine the energy of emitted photon when the atom jumps from the first and third excited states to...
Find the energy required to ionize a ground state hydrogen atom (ie: energy to remove a...
Find the energy required to ionize a ground state hydrogen atom (ie: energy to remove a ground st... (1 bookmark) Find the energy required to ionize a ground state hydrogen atom (ie: energy to remove a ground state electron from an H atom). That is, what is the energy required to make the transition from n = 1 to n = infinity
Consider a trial wavefunction for an electron in H atom in the following form •(r) =...
Consider a trial wavefunction for an electron in H atom in the following form •(r) = re-ar where a is an adjustable parameter. Optimize a so that you obtain the minimum energy (i.e., find the extremum by imposing (E) = 0). How does the minimum energy compare to the ground state energy of an electron? Hint: n! for a>0 A nEN ne-andc= +1 Integration of function f(r, 0,6) in spherical coordinates: $*$*$* $(1,0, 6)r? sin ødødødr f(,0,)ra sin døddr
ANSWER ALL QUESTIONS 1. (a) The hydrogen atom wave functions are written as Unim. State the...
ANSWER ALL QUESTIONS 1. (a) The hydrogen atom wave functions are written as Unim. State the values of n, I, and m. State the relation between a physical quantity and each quantum number. At =0 the hydrogen atom is in the superposition state (7,0) = 4200 + A¥210 V3 where A is a real positive constant. Find A by normalization and determine the wave function at time t > 0. Find the average energy of the electron in eV given...
1) (60 points) The ground state of the hydrogen atom: In three dimensions, the radial part of the Schrodinger equation appropriate for the ground state of the hydrogen atom is given by: ke2 -ħ2 d2 (rR) = E(rR) 2me dr2 where R(r) is a function of r. Here, since we have no angular momentum in the ground state the angular-momentum quantum number /=0. (a) Show that the function R(r) = Ae-Br satisfies the radial Schrodinger equation, and determine the values...
2. The hydrogen atom [8 marks] The time-independent Schrödinger equation for the hydrogen atom in the spherical coordinate representation is where ao-top- 0.5298 10-10rn is the Bohr radius, and μ is the electon-proton reduced mass. Here, the square of the angular momentum operator L2 in the spherical coordinate representation is given by: 2 (2.2) sin θー sin θ 00 The form of the Schrödinger equation means that all energy eigenstates separate into radial and angular motion, and we can write...
In the ground state of the Hydrogen atom the energy of the electron is E state of the He ion? -13.61 eV. What is the energy of the electron in the ground Submit Answer Tries 0/20 What is the energy of the electron in the ground state of the u** ion? Submit Arower Tries 0/20 The electron in the Helon is excited to the n-2 principal state. What is the energy of the electron now? Submit Awer Tries 0/20 What...
45. Use a Gaussian trial function o(r) = e" to compute the ground state energy of the hydrogen atom. What is the "best" value of the variational parameter a? How far off is this energy from the true value?
For the hydrogen atom, its energy at ground state is 13.6 eV, at first excited state is 3.4 eV at second excited state is 1.5 eV and at the third excited state is 0.85 eV. i) Give the energy value for the first two states in Joule (J). [1eV =1.6 x 10-19 J] (2 marks) ii) With the aid of schematic diagram, determine the energy of emitted photon when the atom jumps from the first and third excited states to...
Consider a trial wavefunction for an electron in H atom in the following form •(r) = re-ar where a is an adjustable parameter. Optimize a so that you obtain the minimum energy (i.e., find the extremum by imposing (E) = 0). How does the minimum energy compare to the ground state energy of an electron? Hint: n! for a>0 A nEN ne-andc= +1 Integration of function f(r, 0,6) in spherical coordinates: $*$*$* $(1,0, 6)r? sin ødødødr f(,0,)ra sin døddr
ANSWER ALL QUESTIONS 1. (a) The hydrogen atom wave functions are written as Unim. State the values of n, I, and m. State the relation between a physical quantity and each quantum number. At =0 the hydrogen atom is in the superposition state (7,0) = 4200 + A¥210 V3 where A is a real positive constant. Find A by normalization and determine the wave function at time t > 0. Find the average energy of the electron in eV given...
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