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4. number degeneracies in the When we get to systems with more then 1 quantum of...
4. (20 points) Infinite Wells in Three Dimensions a) Consider a three dimensional in- finite rect...
4. (20 points) Infinite Wells in Three Dimensions a) Consider a three dimensional in- finite rectangular well for which L -L, Ly-2L, ald L2-3L. In terms of quantum numbers (e.g. nz, ny, and n.), M. L, and ћ. write down an expression for the energies of all quantum states. (b) Find the energies of the ground state and the first three lowest lying energies. As in part (b), for each energy level, give the quantum numbers n, ny, n and...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, a...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. Suppose we have an initial state vector lv(t -0) results from Mclntrye without re-deriving them, and you may use a computer for your math as long as you include your code in your solution A(3E1) 4iE2)). You may use E. (4 pts) Use a computer to plot this probability density at 4 times: t 0, t2...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
Griffiths Introductory to Quantum Mechanics (3rd Edition): Problem 7.9 Problem 7.9 Consider a particle of mass...
Griffiths Introductory to Quantum Mechanics (3rd Edition):
Problem 7.9
Problem 7.9 Consider a particle of mass m that is free to move in a one- dimensional region of length L that closes on itself (for instance, a bea that slides frictionlessly on a circular wire of circumference L, as Problem 2.46) (a) Show that the stationary states can be written in the form 2π inx/L where n 0, t l, 2, , and the allowed energies are In Notice that-with...
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force pr...
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force problem (as discussed in TZD Many physical systems involve a particle that moves under the influence of a central force; that is, a force that always points exactly toward, or away from, a force center O. In classical mechanics a famous example of a central force is the force of the sun on a planet. In atomic physics the most obvious example is the hydrogen atom,...
Quantum Mechanics Thank you! 2 Casimir effect We will derive the Casimir effect in three dimensions,...
Quantum Mechanics
Thank you!
2 Casimir effect We will derive the Casimir effect in three dimensions, making use of the Euler- Maclaurin formula Ž 0,F(n) – [F(n)dn = 67\2F'O) + 30 x , F"(0) -... (1) JO n=0 where On = 1 for n > 0, 0 = 1/2, and on = 0 for n < 0. (You don't need to prove this formula.) Let us consider a square box with conducting walls of length L. Let El be the...
Match the following correctly principal quantum number, n=12.3 Al=0, 1, 2, 3, 4 B. designates siz...
Match the following correctly principal quantum number, n=12.3 Al=0, 1, 2, 3, 4 B. designates size and energy C. s and p electrons outside noble gas or angular momentum quantum number, l-0 to (n-1) pseudo-noble gas core, involved in chemical reactions , p, d, f, g-which numbers? magnetic quantum number, m,--l to+1 spin quantum number m s=+1/2 or-1/2 Pauli Exclusion principle Aufbau Principle Hund's Rule pseudo-noble gas core D, no 2 electrons in an atom have the same 4 quantum...
Match the following correctly principal quantum number, n=12.3 Al=0, 1, 2, 3, 4 B. designates size...
Match the following correctly principal quantum number, n=12.3 Al=0, 1, 2, 3, 4 B. designates size and energy C. s and p electrons outside noble gas or angular momentum quantum number, l-0 to (n-1) pseudo-noble gas core, involved in chemical reactions , p, d, f, g-which numbers? magnetic quantum number, m,--l to+1 spin quantum number m s=+1/2 or-1/2 Pauli Exclusion principle Aufbau Principle Hund's Rule pseudo-noble gas core D, no 2 electrons in an atom have the same 4 quantum...
B3 (a) Assume that the T = 0 version of the Fermi-Dirac distribution, namely 1 f...
B3 (a) Assume that the T = 0 version of the Fermi-Dirac distribution, namely 1 f (E) exp [E E)/(kBT) +1 in the usual notation, with Ep the Fermi energy, applies for T> 0. Sketch, on the same axes, the distribution for T = 0 and for T> 0, marking the Fermi energy and indicating the thermal energy kBT 5 Marks (b) In the Sommerfeld model (free electron quantum gas), each electron occupies (n/L)3 of k-space volume. Remembering that we...
TSD.1 In this problem, we will see (in outline) how we can calculate the multiplicity of a monatomic ideal gas This derivation involves concepts presented in chapter 17 Note that the task is to...
TSD.1 In this problem, we will see (in outline) how we can calculate the multiplicity of a monatomic ideal gas This derivation involves concepts presented in chapter 17 Note that the task is to count the number of microstates that are compatible with a given gas macrostate, which we describe by specifying the gas's total energy u (within a tiny range of width dlu), the gas's volume V and the num- ber of molecules N in the gas. We will...
4. (20 points) Infinite Wells in Three Dimensions a) Consider a three dimensional in- finite rectangular well for which L -L, Ly-2L, ald L2-3L. In terms of quantum numbers (e.g. nz, ny, and n.), M. L, and ћ. write down an expression for the energies of all quantum states. (b) Find the energies of the ground state and the first three lowest lying energies. As in part (b), for each energy level, give the quantum numbers n, ny, n and...
A NON stationary state A particle of mass m is in an infinite square well potential of width L, as in McIntyre's section 5.4. Suppose we have an initial state vector lv(t -0) results from Mclntrye without re-deriving them, and you may use a computer for your math as long as you include your code in your solution A(3E1) 4iE2)). You may use E. (4 pts) Use a computer to plot this probability density at 4 times: t 0, t2...
3 Problem Three [10 points] (The Quantum Oscillator) We have seen in class that the Hamiltonian of a particle of a simple Harmonic oscillator potential in one dimension can be expressed in term of the creation and annihilation operators àt and à, respectively, as: or with In >, n = 0,1,..) are the nth eigenstates of the above Hamiltonian. Part A A.1. Show that the energy levels of a simple harmonic oscillator are E,' Aw (nti), n=0, 12, A.2. Calculate...
Griffiths Introductory to Quantum Mechanics (3rd Edition):
Problem 7.9
Problem 7.9 Consider a particle of mass m that is free to move in a one- dimensional region of length L that closes on itself (for instance, a bea that slides frictionlessly on a circular wire of circumference L, as Problem 2.46) (a) Show that the stationary states can be written in the form 2π inx/L where n 0, t l, 2, , and the allowed energies are In Notice that-with...
8.4 The Two-Dimensional Central-Force Problem The 2D harmonic oscillator is a 2D central force problem (as discussed in TZD Many physical systems involve a particle that moves under the influence of a central force; that is, a force that always points exactly toward, or away from, a force center O. In classical mechanics a famous example of a central force is the force of the sun on a planet. In atomic physics the most obvious example is the hydrogen atom,...
Quantum Mechanics
Thank you!
2 Casimir effect We will derive the Casimir effect in three dimensions, making use of the Euler- Maclaurin formula Ž 0,F(n) – [F(n)dn = 67\2F'O) + 30 x , F"(0) -... (1) JO n=0 where On = 1 for n > 0, 0 = 1/2, and on = 0 for n < 0. (You don't need to prove this formula.) Let us consider a square box with conducting walls of length L. Let El be the...
Match the following correctly principal quantum number, n=12.3 Al=0, 1, 2, 3, 4 B. designates size and energy C. s and p electrons outside noble gas or angular momentum quantum number, l-0 to (n-1) pseudo-noble gas core, involved in chemical reactions , p, d, f, g-which numbers? magnetic quantum number, m,--l to+1 spin quantum number m s=+1/2 or-1/2 Pauli Exclusion principle Aufbau Principle Hund's Rule pseudo-noble gas core D, no 2 electrons in an atom have the same 4 quantum...
Match the following correctly principal quantum number, n=12.3 Al=0, 1, 2, 3, 4 B. designates size and energy C. s and p electrons outside noble gas or angular momentum quantum number, l-0 to (n-1) pseudo-noble gas core, involved in chemical reactions , p, d, f, g-which numbers? magnetic quantum number, m,--l to+1 spin quantum number m s=+1/2 or-1/2 Pauli Exclusion principle Aufbau Principle Hund's Rule pseudo-noble gas core D, no 2 electrons in an atom have the same 4 quantum...
B3 (a) Assume that the T = 0 version of the Fermi-Dirac distribution, namely 1 f (E) exp [E E)/(kBT) +1 in the usual notation, with Ep the Fermi energy, applies for T> 0. Sketch, on the same axes, the distribution for T = 0 and for T> 0, marking the Fermi energy and indicating the thermal energy kBT 5 Marks (b) In the Sommerfeld model (free electron quantum gas), each electron occupies (n/L)3 of k-space volume. Remembering that we...
TSD.1 In this problem, we will see (in outline) how we can calculate the multiplicity of a monatomic ideal gas This derivation involves concepts presented in chapter 17 Note that the task is to count the number of microstates that are compatible with a given gas macrostate, which we describe by specifying the gas's total energy u (within a tiny range of width dlu), the gas's volume V and the num- ber of molecules N in the gas. We will...
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