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Quantum Mechanics
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2 Casimir effect We will derive the Casimir effect in three dimensions,...
3 Casimir effect We will derive the Casinir effect in three dimen- sions, making use of the Euler...
3 Casimir effect We will derive the Casinir effect in three dimen- sions, making use of the Euler-Maclaurin formula where θη _ 1 for n > 0,4,-1/2, and θη 0 for n <0. (You don't need to prove this formula.) Let us consider walls of length L. Let EL be the vacuum energy inside the box. We now insert a conducting plate at a distance RL parallel to one of the walls, dividing the volume into two. Denote ER and...
3 Casimir effect c. All vacuum energies considered so far are infinite. In reality, the metal wal...
3 Casimir effect c. All vacuum energies considered so far are infinite. In reality, the metal walls are conduc tors only at finite frequencies, and thus do not impose boundary conditions at infinite frqu cies. (The cutoff is given roughly by the plasma frequencywp Thus, we need (and should) not consider infinite frequencies in our equations. To remove We will derive the Casimir effect in thee d sions, making use of the Euler-Maclaurin formula imen- them, we introduce a cutoff...
(15 points) Encounter with a semi-infinite potential "well" In this problem we will investigate one situation involving a a semi-infinite one-dimensional po- tential well (Figure 1) U=0 regio...
(15 points) Encounter with a semi-infinite potential "well" In this problem we will investigate one situation involving a a semi-infinite one-dimensional po- tential well (Figure 1) U=0 region 1 region 2 region 3 Figure 1: Semi-infinite potential for Problem 3 This potential is piecewise defined as follows where Uo is some positive value of energy. The three intervals in x have been labeled region 1,2 and 3 in Figure 1 Consider a particle of mass m f 0 moving in...
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...
Question 8 please 5. We start with Schrodinger's Equation in 2(x,t) = H¥(x,t). We can write...
Question 8 please
5. We start with Schrodinger's Equation in 2(x,t) = H¥(x,t). We can write the time derivative as 2.4(x, t) = V(x,+) - (xt), where At is a sufficiently small increment of time. Plug the algebraic form of the derivative into Schrodinger's Eq. and solve for '(x,t+At). b. Put your answer in the form (x,t+At) = T '(x,t). c. What physically does the operator T do to the function '(x,t)? d. Deduce an expression for '(x,t+24t), in terms...
12 2. Consider the heat equation where for simplicity we take c = 1. Thus au...
12 2. Consider the heat equation where for simplicity we take c = 1. Thus au du ar2 at Suppose that a heat conducting rod of length a has the left end r = ( maintained at temperature ( while the right end at r = is insulated so that there is no heat flow. This gives us the boundary conditions au u(0,t) = 0, (7,0) = 0. Find the solution u(x, t) if the initial temperature distribution on the...
ANSWER 2 & 3 please. Show work for my understanding and upvote. THANK YOU!! 2. Given a regular n-gon, let r be a rotation of it by 2π/n radians. This time, assume that we are not allowed to flip...
ANSWER 2 & 3 please. Show work for my understanding and
upvote. THANK YOU!!
2. Given a regular n-gon, let r be a rotation of it by 2π/n radians. This time, assume that we are not allowed to flip over the n-gon. These n actions form a group denotecd (a) Draw a Cayley diagram for Cn for n-4, n-5, and n-6 (b) For n 4, 5, 6, find all minimal generating sets of C.· [Note: There are minimal generating sets...
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...
Question 2: finite square well in three dimensions 12 marks *Please note: in PHYS2111 we have...
Question 2: finite square well in three dimensions 12 marks *Please note: in PHYS2111 we have not discussed multi-dimensional systems, but please keep in mind that in order to answer this question all you need is the knowledge about a particle moving in one dimension in a finite square well. Consider a particle of mass m moving in a three-dimensional spherically symmetric square-well potential of radius a and depth V. (see also figure on pag. 3): V(r) = { S-Vo...
some useful examples the 1st one is the question where x is a vector while the second are examples. (d) Now consider the N-dimensional vector x an the integral ((22%)lejbTA-lb. (23.46)...
some useful examples
the 1st one is the question where x is a vector while
the second are examples.
(d) Now consider the N-dimensional vector x an the integral ((22%)lejbTA-lb. (23.46) det A By differentiating with respect to components of the vector b, and then setting b 0, show that (r,a) (23.47) (e) Using these results, argue that +(A)u(A (23.48) j k. (f) Write down an expression for the general case Ti.z) This is the basis of Wick's theorem in...
3 Casimir effect We will derive the Casinir effect in three dimen- sions, making use of the Euler-Maclaurin formula where θη _ 1 for n > 0,4,-1/2, and θη 0 for n <0. (You don't need to prove this formula.) Let us consider walls of length L. Let EL be the vacuum energy inside the box. We now insert a conducting plate at a distance RL parallel to one of the walls, dividing the volume into two. Denote ER and...
3 Casimir effect c. All vacuum energies considered so far are infinite. In reality, the metal walls are conduc tors only at finite frequencies, and thus do not impose boundary conditions at infinite frqu cies. (The cutoff is given roughly by the plasma frequencywp Thus, we need (and should) not consider infinite frequencies in our equations. To remove We will derive the Casimir effect in thee d sions, making use of the Euler-Maclaurin formula imen- them, we introduce a cutoff...
(15 points) Encounter with a semi-infinite potential "well" In this problem we will investigate one situation involving a a semi-infinite one-dimensional po- tential well (Figure 1) U=0 region 1 region 2 region 3 Figure 1: Semi-infinite potential for Problem 3 This potential is piecewise defined as follows where Uo is some positive value of energy. The three intervals in x have been labeled region 1,2 and 3 in Figure 1 Consider a particle of mass m f 0 moving in...
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...
Question 8 please
5. We start with Schrodinger's Equation in 2(x,t) = H¥(x,t). We can write the time derivative as 2.4(x, t) = V(x,+) - (xt), where At is a sufficiently small increment of time. Plug the algebraic form of the derivative into Schrodinger's Eq. and solve for '(x,t+At). b. Put your answer in the form (x,t+At) = T '(x,t). c. What physically does the operator T do to the function '(x,t)? d. Deduce an expression for '(x,t+24t), in terms...
12 2. Consider the heat equation where for simplicity we take c = 1. Thus au du ar2 at Suppose that a heat conducting rod of length a has the left end r = ( maintained at temperature ( while the right end at r = is insulated so that there is no heat flow. This gives us the boundary conditions au u(0,t) = 0, (7,0) = 0. Find the solution u(x, t) if the initial temperature distribution on the...
ANSWER 2 & 3 please. Show work for my understanding and
upvote. THANK YOU!!
2. Given a regular n-gon, let r be a rotation of it by 2π/n radians. This time, assume that we are not allowed to flip over the n-gon. These n actions form a group denotecd (a) Draw a Cayley diagram for Cn for n-4, n-5, and n-6 (b) For n 4, 5, 6, find all minimal generating sets of C.· [Note: There are minimal generating sets...
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...
Question 2: finite square well in three dimensions 12 marks *Please note: in PHYS2111 we have not discussed multi-dimensional systems, but please keep in mind that in order to answer this question all you need is the knowledge about a particle moving in one dimension in a finite square well. Consider a particle of mass m moving in a three-dimensional spherically symmetric square-well potential of radius a and depth V. (see also figure on pag. 3): V(r) = { S-Vo...
some useful examples
the 1st one is the question where x is a vector while
the second are examples.
(d) Now consider the N-dimensional vector x an the integral ((22%)lejbTA-lb. (23.46) det A By differentiating with respect to components of the vector b, and then setting b 0, show that (r,a) (23.47) (e) Using these results, argue that +(A)u(A (23.48) j k. (f) Write down an expression for the general case Ti.z) This is the basis of Wick's theorem in...
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