Show that the expectation values of the three components of S in the state χ1 2...
Show that the expectation values of the three components of S in the state χ1 2 are hSxi = h¯ 2 sin θ cos φ (8) hSyi = h¯ 2 sin θ sin φ (9) hSzi = h¯ 2 cos θ (10) This result shows that the direction of the vector hSi is given by the same polar angles (θ, φ) which define ˆn. In this sense, we can say that χ1 2 is the the state of spin “up” in the direction defined by ˆn. (e) Compute now the expectation value of hSi in the state χ− 1 2 , and
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Show that the expectation values of the three components of S in the state χ1 2...
2. Spin-1/2 system: (20 points) The Pauli matrices are, 0 -1 from which we can define...
2. Spin-1/2 system: (20 points) The Pauli matrices are, 0 -1 from which we can define the spin matrices, s.-슬&z, Šv = , S.-출.. We'll use the eigenkets of S that, for the spin half system, they can be represented by the spinors, a) Show, by matrix multiplication that |+) and |-) are eigenstates of the S operator and determine the eigenvalues. Show that they are not eigenstates of S and Sy b) Show that the matrix squares s ,...
2- If the z-component of an electron spin is +h/2, what is the probability that its component along a direction z', that forms an angle θ with the z-axis, equals +h/2 or-h/2? What is the avera...
2- If the z-component of an electron spin is +h/2, what is the probability that its component along a direction z', that forms an angle θ with the z-axis, equals +h/2 or-h/2? What is the average value of the spin along z'? (Hint. Sz.-S. n where n; sin θ cospi + sin θ sin φ j + cos θ k is a unit vector along z'.) (10 Scores)
2- If the z-component of an electron spin is +h/2, what is...
I. The state vector φ(t) at time t can be decomposed on the {I +), l-)} basis: Write down the sys...
I. The state vector φ(t) at time t can be decomposed on the {I +), l-)} basis: Write down the system of coupled differential equations which the components c+ (t) and c() satisfy 2. Let |φ(1-0)) be decomposed on the {lx+), lx-) basis. Show that c+(1)-(Ηφ(t)) is written as sin with Ω 2VA2+B2. Here f.Ω is the energy difference of the two levels. Show that c+(1) (as well as c (t)) satisfies the differential equation c) 0. We assume that...
4.8. A spin- particle, initially in a state with S h/2 with n sin i+ cos...
4.8. A spin- particle, initially in a state with S h/2 with n sin i+ cos k, is in a constant magnetic field Bo in the z direction. Determine the state of the particle at time and determine how (S,), (S), and (S.) vary with time.
Problem 2. (30 points) The spin states: s 1,m) and Is -2, m1) composed of spin-3/2 and spin-1/2 states are linear combi...
Problem 2. (30 points) The spin states: s 1,m) and Is -2, m1) composed of spin-3/2 and spin-1/2 states are linear combinations of s1 3/2,m-3/2;2 1/2,m2 1/2) and 81-3/2, m-1/2; 2 1/2, m2--1/2), that is 11.-1)-cos θ3/2,-3/2; 1/2, 1/2) _ sin θ|3/2.-1/2; 1/2,-1/2), 2.-1) sin θ|3/2,-3/2; 1/2, 1/2) + cos θ|3/2.-1/2: 1/2,-1/2) a) Determine the values for cos θ and sin θ b) Express |3/2,-3/2; 1/2, 1/2) and |3/2,-1/2;1/2,-1/2) as functions of |1, -1) and 2,-1) c) A system of...
1. This quasi-"walkthrough" problem is great practice in cross-products, vectors, and integration. Consider the current loop shown in Figure 2, with B _Bx, and the loop lying in the x - z pl...
1. This quasi-"walkthrough" problem is great practice in cross-products, vectors, and integration. Consider the current loop shown in Figure 2, with B _Bx, and the loop lying in the x - z plane of the page (y points into the page). We wish to find the net torque on this current loop. we'll do this by integrating in θ, the angle shown (a) We'll start with a little segment dl at point p as shown in the figure. What is...
g vluelty 1im/s,0,-2m/s), and its mass is 0.5kg? sr=13 m, 5 m,--2 m Consider a rider...
g vluelty 1im/s,0,-2m/s), and its mass is 0.5kg? sr=13 m, 5 m,--2 m Consider a rider on a Ferris wheel of radius R that s in the xz plane and is rotating clockwise with angu- lar velocity W. At every instant, then, a person riding or ider of mass m who at a certain instant is passing the (a) At that instant, what is the magnitude and direction of (b) What is the magnitude and direction of the rider's wheel's...
A. Make a sketch of a vector F- (x,y, z), labeling the appropriate spherical coordinates. In addi...
A. Make a sketch of a vector F- (x,y, z), labeling the appropriate spherical coordinates. In addition, show the unit vectors r, θ, and φ at that point B. Write the vectors ŕ.0, and ф in terms of the unit vectors x, y, and г. Here's the easy way to do this 1. For r, simply use the fact that/r 2. For φ, use the following formula sin θ Explain why the above formula works 3. Compute θ via θ...
#1 Exercises with Vectors-II Name [1] Suppose you have two vectors, a and b, that have...
#1
Exercises with Vectors-II Name [1] Suppose you have two vectors, a and b, that have the same length, so that lal-lb but they point in different directions. Denote the angle between them by . Show that tan(0/2) la-bMa+b Hint: Compute the right-hand side using the fact that lal-bl, and the trig identities 1-cos θ-2sin'(9/2) and 1+cos9-2cos(θ/2) 12] Vectors in 3-dimensions are often parameterized in terms of their length and two angles, as shown in the figure (think of a...
B.2. The surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r,...
B.2. The surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r, 0, 0) by where θ0 is a constant (which you may assume is less than π/2) (a) Sketch the surface Sc (b) Using the expression show that the vector element of area on Sc is given by -T Sin where [41 (c) The vector field a(r) is given in Cartesian coordinates by Show that on Sc and hence that 4 2 (d) The curved...
2. Spin-1/2 system: (20 points) The Pauli matrices are, 0 -1 from which we can define the spin matrices, s.-슬&z, Šv = , S.-출.. We'll use the eigenkets of S that, for the spin half system, they can be represented by the spinors, a) Show, by matrix multiplication that |+) and |-) are eigenstates of the S operator and determine the eigenvalues. Show that they are not eigenstates of S and Sy b) Show that the matrix squares s ,...
2- If the z-component of an electron spin is +h/2, what is the probability that its component along a direction z', that forms an angle θ with the z-axis, equals +h/2 or-h/2? What is the average value of the spin along z'? (Hint. Sz.-S. n where n; sin θ cospi + sin θ sin φ j + cos θ k is a unit vector along z'.) (10 Scores)
2- If the z-component of an electron spin is +h/2, what is...
I. The state vector φ(t) at time t can be decomposed on the {I +), l-)} basis: Write down the system of coupled differential equations which the components c+ (t) and c() satisfy 2. Let |φ(1-0)) be decomposed on the {lx+), lx-) basis. Show that c+(1)-(Ηφ(t)) is written as sin with Ω 2VA2+B2. Here f.Ω is the energy difference of the two levels. Show that c+(1) (as well as c (t)) satisfies the differential equation c) 0. We assume that...
4.8. A spin- particle, initially in a state with S h/2 with n sin i+ cos k, is in a constant magnetic field Bo in the z direction. Determine the state of the particle at time and determine how (S,), (S), and (S.) vary with time.
Problem 2. (30 points) The spin states: s 1,m) and Is -2, m1) composed of spin-3/2 and spin-1/2 states are linear combinations of s1 3/2,m-3/2;2 1/2,m2 1/2) and 81-3/2, m-1/2; 2 1/2, m2--1/2), that is 11.-1)-cos θ3/2,-3/2; 1/2, 1/2) _ sin θ|3/2.-1/2; 1/2,-1/2), 2.-1) sin θ|3/2,-3/2; 1/2, 1/2) + cos θ|3/2.-1/2: 1/2,-1/2) a) Determine the values for cos θ and sin θ b) Express |3/2,-3/2; 1/2, 1/2) and |3/2,-1/2;1/2,-1/2) as functions of |1, -1) and 2,-1) c) A system of...
1. This quasi-"walkthrough" problem is great practice in cross-products, vectors, and integration. Consider the current loop shown in Figure 2, with B _Bx, and the loop lying in the x - z plane of the page (y points into the page). We wish to find the net torque on this current loop. we'll do this by integrating in θ, the angle shown (a) We'll start with a little segment dl at point p as shown in the figure. What is...
g vluelty 1im/s,0,-2m/s), and its mass is 0.5kg? sr=13 m, 5 m,--2 m Consider a rider on a Ferris wheel of radius R that s in the xz plane and is rotating clockwise with angu- lar velocity W. At every instant, then, a person riding or ider of mass m who at a certain instant is passing the (a) At that instant, what is the magnitude and direction of (b) What is the magnitude and direction of the rider's wheel's...
A. Make a sketch of a vector F- (x,y, z), labeling the appropriate spherical coordinates. In addition, show the unit vectors r, θ, and φ at that point B. Write the vectors ŕ.0, and ф in terms of the unit vectors x, y, and г. Here's the easy way to do this 1. For r, simply use the fact that/r 2. For φ, use the following formula sin θ Explain why the above formula works 3. Compute θ via θ...
#1
Exercises with Vectors-II Name [1] Suppose you have two vectors, a and b, that have the same length, so that lal-lb but they point in different directions. Denote the angle between them by . Show that tan(0/2) la-bMa+b Hint: Compute the right-hand side using the fact that lal-bl, and the trig identities 1-cos θ-2sin'(9/2) and 1+cos9-2cos(θ/2) 12] Vectors in 3-dimensions are often parameterized in terms of their length and two angles, as shown in the figure (think of a...
B.2. The surface Sc of an ice-cream cone can be parametrised in spherical polar coordinates (r, 0, 0) by where θ0 is a constant (which you may assume is less than π/2) (a) Sketch the surface Sc (b) Using the expression show that the vector element of area on Sc is given by -T Sin where [41 (c) The vector field a(r) is given in Cartesian coordinates by Show that on Sc and hence that 4 2 (d) The curved...
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