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(VI.1) At timet 0, the 3D wavefunction of a particle is (x, y, 2) A(x t...
Probability. A wavefunction ψ(x) describing the state of a particle free to move along one dimens...
Probability. A wavefunction ψ(x) describing the state of a particle free to move along one dimension x is given by the following expression: x <0,x>2L (A) Determine the value of the normalization constant c. (B) Draw the wavefunction. (C) Calculate Prob(L/2 S x 3 3L/2), the probability of finding the particle between x - L/2 and 3L/2
Probability. A wavefunction ψ(x) describing the state of a particle free to move along one dimension x is given by the following expression:...
4. The wavefunction of a particle at t = 0) is given by: 4(x,0) = Cexp(...
4. The wavefunction of a particle at t = 0) is given by: 4(x,0) = Cexp( ), Xo = real constant (a) Sketch the wavefunction and normalise it to find |C|. (The sketch should help to suggest how to select the method of integration.) (b) Determine the probability of finding the particle at a value of x between –a and a, ie. -a < x < a. [7] [8]
4 At a given time, the wave function of a particle of mass m moving in...
4 At a given time, the wave function of a particle of mass m moving in a potential well is constant}(xyz) exp(-aa real a > 0. Calculate the probabilities that measurements of L2 and L,z yield the results 2h2 and 0 respectively. [6]
4 At a given time, the wave function of a particle of mass m moving in a potential well is constant}(xyz) exp(-aa real a > 0. Calculate the probabilities that measurements of L2 and L,z yield the...
Recall that the time evolution of a wavefunction y(x, t) is determined by the Schrödinger equation,...
Recall that the time evolution of a wavefunction y(x, t) is determined by the Schrödinger equation, which in position space reads iħ 4(x, t) = -24(x, t) + V(x, t)(x, t). ih vrt - h ? a) Consider any two normalized solutions to the Schrödinger equation, 41(2, t) and 02(3,t). Prove that their inner product is independent of time, doo 1 Vi (2, t)u2(x, t) dc = 0. dt J-00 Hint: prove the useful intermediate result, a 202 201 -...
P7B.8 A normalized wavefunction for a particle confined between 0 and L in the x direction,...
P7B.8 A normalized wavefunction for a particle confined between 0 and L in the x direction, and between 0 and L in the y direction (that is, to a square of side L) is Ψ= (2/L) sin(nx/L) sin(ny/L). The probability of finding the particle between x, and x, along x, and between y, and y, along y is P- Calculate the probability that the particle is: (a) between 0 and x L/2,y O and y L/2 (i.e, in the bottom...
322 CHAPTER 5. ANGULAR MOMENTUM Problem 5.12 Consider a particle whose wave function is 1 222-x2-y2...
322 CHAPTER 5. ANGULAR MOMENTUM Problem 5.12 Consider a particle whose wave function is 1 222-x2-y2 4 A 3 xz (x, y, z) = 2 2 Calculate L2 (x, y, z) and L-y(x, y, z). Find the total angular momentum of this particle. (b) Calculate L+ y (x, y, z) and (Y L+ W). (c)If a measurement of the z-component of the orbital angular momentum is carried out, find the probabilities corresponding to finding the results 0, h, and -h....
For timet > 0, the acceleration of a particle moving along the x-axis is given by...
For timet > 0, the acceleration of a particle moving along the x-axis is given by a (t) = cost+t. At time t = 0, the velocity of the particle is 8 and the position of the particle is 0. What is the position of the particle at time t = ?
4 Spherical harmonics in action Consider a particle in three dimensions in a state (2, y,...
4 Spherical harmonics in action Consider a particle in three dimensions in a state (2, y, z) = A2 + y + 2z)e-ar where a > 0 is a constant and A is a normalization factor. (a) Write the spherical harmonics Y0+ in terms of x,y,z, and r. (b) Using your result from part (a), show that can be written as a product of a purely r-dependent wavefunction and a linear combination of spherical harmonics.
5. The function x< 0 0 < x < a ψ(x)-Ax(1-(x/a)] is an acceptable wavefunction for...
5. The function x< 0 0 < x < a ψ(x)-Ax(1-(x/a)] is an acceptable wavefunction for a particle in a one-dimensional space (x can take values between -oo and +oo) (a) Give two reasons why this is an acceptable wave function. (b) Calculate the normalization constant A. (c) Using the definition for the average of an observable "o" described by the operator "o": and to)
3. Consider a particle of mass m moving in a potential given by: W (2, y,...
3. Consider a particle of mass m moving in a potential given by: W (2, y, z) = 0 < x <a,0 < y <a l+o, elsewhere a) Write down the total energy and the 3D wavefunction for this particle. b) Assuming that hw > 312 h2/(2ma), find the energies and the corresponding degen- eracies for the ground state and the first excited state. c) Assume now that, in addition to the potential V(x, y, z), this particle also has...
Probability. A wavefunction ψ(x) describing the state of a particle free to move along one dimension x is given by the following expression: x <0,x>2L (A) Determine the value of the normalization constant c. (B) Draw the wavefunction. (C) Calculate Prob(L/2 S x 3 3L/2), the probability of finding the particle between x - L/2 and 3L/2
Probability. A wavefunction ψ(x) describing the state of a particle free to move along one dimension x is given by the following expression:...
4. The wavefunction of a particle at t = 0) is given by: 4(x,0) = Cexp( ), Xo = real constant (a) Sketch the wavefunction and normalise it to find |C|. (The sketch should help to suggest how to select the method of integration.) (b) Determine the probability of finding the particle at a value of x between –a and a, ie. -a < x < a. [7] [8]
4 At a given time, the wave function of a particle of mass m moving in a potential well is constant}(xyz) exp(-aa real a > 0. Calculate the probabilities that measurements of L2 and L,z yield the results 2h2 and 0 respectively. [6]
4 At a given time, the wave function of a particle of mass m moving in a potential well is constant}(xyz) exp(-aa real a > 0. Calculate the probabilities that measurements of L2 and L,z yield the...
Recall that the time evolution of a wavefunction y(x, t) is determined by the Schrödinger equation, which in position space reads iħ 4(x, t) = -24(x, t) + V(x, t)(x, t). ih vrt - h ? a) Consider any two normalized solutions to the Schrödinger equation, 41(2, t) and 02(3,t). Prove that their inner product is independent of time, doo 1 Vi (2, t)u2(x, t) dc = 0. dt J-00 Hint: prove the useful intermediate result, a 202 201 -...
P7B.8 A normalized wavefunction for a particle confined between 0 and L in the x direction, and between 0 and L in the y direction (that is, to a square of side L) is Ψ= (2/L) sin(nx/L) sin(ny/L). The probability of finding the particle between x, and x, along x, and between y, and y, along y is P- Calculate the probability that the particle is: (a) between 0 and x L/2,y O and y L/2 (i.e, in the bottom...
322 CHAPTER 5. ANGULAR MOMENTUM Problem 5.12 Consider a particle whose wave function is 1 222-x2-y2 4 A 3 xz (x, y, z) = 2 2 Calculate L2 (x, y, z) and L-y(x, y, z). Find the total angular momentum of this particle. (b) Calculate L+ y (x, y, z) and (Y L+ W). (c)If a measurement of the z-component of the orbital angular momentum is carried out, find the probabilities corresponding to finding the results 0, h, and -h....
For timet > 0, the acceleration of a particle moving along the x-axis is given by a (t) = cost+t. At time t = 0, the velocity of the particle is 8 and the position of the particle is 0. What is the position of the particle at time t = ?
4 Spherical harmonics in action Consider a particle in three dimensions in a state (2, y, z) = A2 + y + 2z)e-ar where a > 0 is a constant and A is a normalization factor. (a) Write the spherical harmonics Y0+ in terms of x,y,z, and r. (b) Using your result from part (a), show that can be written as a product of a purely r-dependent wavefunction and a linear combination of spherical harmonics.
5. The function x< 0 0 < x < a ψ(x)-Ax(1-(x/a)] is an acceptable wavefunction for a particle in a one-dimensional space (x can take values between -oo and +oo) (a) Give two reasons why this is an acceptable wave function. (b) Calculate the normalization constant A. (c) Using the definition for the average of an observable "o" described by the operator "o": and to)
3. Consider a particle of mass m moving in a potential given by: W (2, y, z) = 0 < x <a,0 < y <a l+o, elsewhere a) Write down the total energy and the 3D wavefunction for this particle. b) Assuming that hw > 312 h2/(2ma), find the energies and the corresponding degen- eracies for the ground state and the first excited state. c) Assume now that, in addition to the potential V(x, y, z), this particle also has...
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