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Exercise 7: Variational principle and hydrogen atom a) Variational method: show that EOT OTHOT)orloT) yields an upper b...
2. The unational method is an incredibly simple but surprisingly powerful method for understanding the low- energy beha...
2. The unational method is an incredibly simple but surprisingly powerful method for understanding the low- energy behavior of quantu systems. It is used constantly in marny-body physics and in quantum chemistry. The main idea is thst for any physical Hamiltonian, there is a lowest energy state, i.e. the ground state Ipo). All other states (ignoring degeneracy) have higher energy that this one. Therefore we have Therefore, to get an upper bound on the energy of Eo, it suffices to...
Please try to write legibly 3. (20 pts) Variational Principle Find an upper bound on the...
Please try to write legibly
3. (20 pts) Variational Principle Find an upper bound on the ground state energy of the Hamiltonian ölusing the trial wavefunctions »-Lesol cOS eBx s-e 1
(100pts) For full credit show ALL work. The variational method is commonly used in various guises...
(100pts) For full credit show ALL work. The variational method is commonly used in various guises in physics to determine the quantum ground states of solids. One of the properties of ground states is that they are local minima with respect to some parameter (in this case μ ) thus they are the "lowest energy state" one can obtain as a function of this parameter. Consider a "hydrogen like" atom. Assume the electron is n the (r)-Be-u state, where B...
a) Use the variational method to estimate the binding energy of a deuteron. Assume that the...
a) Use the variational method to estimate the binding energy of a deuteron. Assume that the potential between the proton and neutron is V(r) = Ae-r/ro where A and ro are constants and use as a trial function W(r) = Ce-Br (4) where C is the normalization constant. b) Consider the Hamiltonian of a nonharmonic oscillator d2 (5) H + x2 + x4 dx2 Use the WKB approximation to find the ground state of the system as x .
Two students have a very pressing homework deadline concerning the application of the variational principle to...
Two students have a very pressing homework deadline concerning the application of the variational principle to estimate the ground state energy of the harmonic oscillator. The Hamiltonian operator of such system is î H -12d = 24 d.22 + 2 .2. in which u is the reduced mass of the oscillator and w = (force constant/u)/2 its natural frequency. The correct energies for this system are well known Eo = (v +) , v= 0,1,2, ... As the trial function...
2. Variational method. We can approximate the true ground-state wavefunction of the harmonic -프sxs and w(x) =D0 cos(cx...
2. Variational method. We can approximate the true ground-state wavefunction of the harmonic -프sxs and w(x) =D0 cos(cx) in the range oscillator by the trial wavefunction p(x) = X 2c 2c outside this range. (This wavefunction is already normalized). (A) Compute the energy expectation value of b as a function of c. (B) Determine the value of c that gives the minimal energy. (C) Compare the minimal energy to the energy of the true ground-state wavefunction
2. Variational method. We...
11.17 A positronium atom is a hydrogen-like atom with a positron (m = meq = te,...
11.17 A positronium atom is a hydrogen-like atom with a positron (m = meq = te, spin 1/2) as a nucleus and a bound electron. The hyperfine structure in the ground state of positronium is described by a perturbation Hamiltonian H' = AS, S/hwhere S, are the spins of the elec- tron and positron. a) What is the Bohr energy of the ground state of positronium (ignore hyperfine structure for now)? b) The electron and positron spins can be coupled...
Take as a a trial function for the ground state of the hydrogen atom (a) e^(−kr),...
Take as a a trial function for the ground state of the hydrogen atom (a) e^(−kr), (b) e^(−kr^2) and use the variational principle to find the optimal value of k in each case. Identify the better wavefunction. The only part of the laplacian that need be considered is the part that involves radial derivatives. Please show your work and explain your answer so I can follow along yes it's not organic chemistry, its quantum mechanics (chemistry)
Problem 7.49 A hydrogen atom is placed in a uniform magnetic field Bo Bo (the Hamiltonian can be ...
Problem 7.49
Problem 7.49 A hydrogen atom is placed in a uniform magnetic field Bo Bo (the Hamiltonian can be written as in Equation 4.230). Use the Feynman-Hellman theorem (Problem 7.38) to show that a En (7.114) where the electron's magnetic dipole moment10 (orbital plus spin) is Yo l-mechanical + γ S . μ The mechanical angular momentum is defined in Equation 4.231 a volume V and at 0 K (when they're all in the ground state) is41 Note: From...
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...
2. The unational method is an incredibly simple but surprisingly powerful method for understanding the low- energy behavior of quantu systems. It is used constantly in marny-body physics and in quantum chemistry. The main idea is thst for any physical Hamiltonian, there is a lowest energy state, i.e. the ground state Ipo). All other states (ignoring degeneracy) have higher energy that this one. Therefore we have Therefore, to get an upper bound on the energy of Eo, it suffices to...
Please try to write legibly
3. (20 pts) Variational Principle Find an upper bound on the ground state energy of the Hamiltonian ölusing the trial wavefunctions »-Lesol cOS eBx s-e 1
(100pts) For full credit show ALL work. The variational method is commonly used in various guises in physics to determine the quantum ground states of solids. One of the properties of ground states is that they are local minima with respect to some parameter (in this case μ ) thus they are the "lowest energy state" one can obtain as a function of this parameter. Consider a "hydrogen like" atom. Assume the electron is n the (r)-Be-u state, where B...
a) Use the variational method to estimate the binding energy of a deuteron. Assume that the potential between the proton and neutron is V(r) = Ae-r/ro where A and ro are constants and use as a trial function W(r) = Ce-Br (4) where C is the normalization constant. b) Consider the Hamiltonian of a nonharmonic oscillator d2 (5) H + x2 + x4 dx2 Use the WKB approximation to find the ground state of the system as x .
Two students have a very pressing homework deadline concerning the application of the variational principle to estimate the ground state energy of the harmonic oscillator. The Hamiltonian operator of such system is î H -12d = 24 d.22 + 2 .2. in which u is the reduced mass of the oscillator and w = (force constant/u)/2 its natural frequency. The correct energies for this system are well known Eo = (v +) , v= 0,1,2, ... As the trial function...
2. Variational method. We can approximate the true ground-state wavefunction of the harmonic -프sxs and w(x) =D0 cos(cx) in the range oscillator by the trial wavefunction p(x) = X 2c 2c outside this range. (This wavefunction is already normalized). (A) Compute the energy expectation value of b as a function of c. (B) Determine the value of c that gives the minimal energy. (C) Compare the minimal energy to the energy of the true ground-state wavefunction
2. Variational method. We...
11.17 A positronium atom is a hydrogen-like atom with a positron (m = meq = te, spin 1/2) as a nucleus and a bound electron. The hyperfine structure in the ground state of positronium is described by a perturbation Hamiltonian H' = AS, S/hwhere S, are the spins of the elec- tron and positron. a) What is the Bohr energy of the ground state of positronium (ignore hyperfine structure for now)? b) The electron and positron spins can be coupled...
Problem 7.49
Problem 7.49 A hydrogen atom is placed in a uniform magnetic field Bo Bo (the Hamiltonian can be written as in Equation 4.230). Use the Feynman-Hellman theorem (Problem 7.38) to show that a En (7.114) where the electron's magnetic dipole moment10 (orbital plus spin) is Yo l-mechanical + γ S . μ The mechanical angular momentum is defined in Equation 4.231 a volume V and at 0 K (when they're all in the ground state) is41 Note: From...
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...
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