Homework Answers
The energy eigen value equation -
H|n,l,m>=En|n,l,m>
Where En=-13.6/n2n2(ev)
H|n,l,m>=(-13.5ev/n2)|n,l,m>
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6. For the hydrogen atom system with state In, l,m) and H is its Hamiltonian, HIn,...
1. Given a state y(r) expanded on the eigenstates of the Hamiltonian for the electron, H, in a hydrogen atom: where the...
1. Given a state y(r) expanded on the eigenstates of the Hamiltonian for the electron, H, in a hydrogen atom: where the subscript of E is n, the principal quantum number. The other two numbers are the 1 and m values, find the expectation values of H (you may use the eigenvalue equation to evaluate for H), L-(total angular momentum operator square), Lz (the z-component of the angular momentum operator) and P (parity operator). Draw schematic pictures of 1 and...
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...
Consider a hydrogen atom in its ground state. The hyperfine interaction between the magnetic moment of...
Consider a hydrogen atom in its ground state. The hyperfine interaction between the magnetic moment of the proton and the magnetic moment of the electron is described by the Hamiltonian: H =A S.S, where S, is the spin of the electron, S, is the spin of the proton, and A is a constant. The ground state is split by the hyperfine coupling. Obtain the energies of the split levels.
A hydrogen atom is in its fifth excited state, with principal quantum number 6. The atom...
A hydrogen atom is in its fifth excited state, with principal
quantum number 6. The atom emits a photon with a wavelength of 1
096 nm. Determine the maximum possible magnitude of the orbital
angular momentum of the atom after emission.
A hydrogen atom·s in its fifth excited state, with principal quantum number 6 The atom emits a photon with a wavelength of 1 096 nm. Determine the maximum possible magnitude of the orbital angular momentum of the atom after...
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...
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...
The Hamiltonian of a system in the basis In > is given by H = hw("...
The Hamiltonian of a system in the basis In > is given by H = hw(" >< 0,1 + il" >< 421-142 >< 0,1 -21°3 >< $3D Here w is a constant. Write the Hamiltonian in the form of a matrix and obtain its eigenvalues and eigenfunctions. Express the eigenfunctions in terms of the basis In > and in its eigenvalues as En = hwe If the system is initially in the state | (0) >= 10 > a. What...
6. (20 points) Consider a free particle of mass m in a cubical box of side L with the Hamiltonian H = - -V2. We assume...
6. (20 points) Consider a free particle of mass m in a cubical box of side L with the Hamiltonian H = - -V2. We assume periodic boundary condition (a) Find the eigenfunctions (F) and its eigenvalue E (b) In the coordinate representation, find the density matrix in the canonical ensemble That is, to find (7le-BH) (c) Find the trace of the density matrix.
6. (20 points) Consider a free particle of mass m in a cubical box of side...
1. Given a state y(r) expanded on the eigenstates of the Hamiltonian for the electron, H, in a hydrogen atom: where the subscript of E is n, the principal quantum number. The other two numbers are the 1 and m values, find the expectation values of H (you may use the eigenvalue equation to evaluate for H), L-(total angular momentum operator square), Lz (the z-component of the angular momentum operator) and P (parity operator). Draw schematic pictures of 1 and...
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...
Consider a hydrogen atom in its ground state. The hyperfine interaction between the magnetic moment of the proton and the magnetic moment of the electron is described by the Hamiltonian: H =A S.S, where S, is the spin of the electron, S, is the spin of the proton, and A is a constant. The ground state is split by the hyperfine coupling. Obtain the energies of the split levels.
A hydrogen atom is in its fifth excited state, with principal
quantum number 6. The atom emits a photon with a wavelength of 1
096 nm. Determine the maximum possible magnitude of the orbital
angular momentum of the atom after emission.
A hydrogen atom·s in its fifth excited state, with principal quantum number 6 The atom emits a photon with a wavelength of 1 096 nm. Determine the maximum possible magnitude of the orbital angular momentum of the atom after...
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...
The Hamiltonian of a system in the basis In > is given by H = hw(" >< 0,1 + il" >< 421-142 >< 0,1 -21°3 >< $3D Here w is a constant. Write the Hamiltonian in the form of a matrix and obtain its eigenvalues and eigenfunctions. Express the eigenfunctions in terms of the basis In > and in its eigenvalues as En = hwe If the system is initially in the state | (0) >= 10 > a. What...
6. (20 points) Consider a free particle of mass m in a cubical box of side L with the Hamiltonian H = - -V2. We assume periodic boundary condition (a) Find the eigenfunctions (F) and its eigenvalue E (b) In the coordinate representation, find the density matrix in the canonical ensemble That is, to find (7le-BH) (c) Find the trace of the density matrix.
6. (20 points) Consider a free particle of mass m in a cubical box of side...
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