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Problem # 5 ( 6 pts) Consider the potential: V(x) αδ(x) Where α is a positive...
1. A particle is scattered upon by the potential V(x)-re(x). (β is a positive constant) (a)...
1. A particle is scattered upon by the potential V(x)-re(x). (β is a positive constant) (a) Write down the proper boundary conditions. (b) Find the reflection and transmission coefficients for E>0. (20%)
5. Consider a square potential barrier in figure below: V(x) 0 x <0 a) Assume that...
5. Consider a square potential barrier in figure below: V(x) 0 x <0 a) Assume that incident particles of energy E> v are coming from-X. Find the stationary states (the equations for region . 2 and 3 and the main equation for the all regions). Apply the matching limit conditions in the figure. Explain and find all the constants used in the equations in terms of the parameters provided and Planck's constant -(6) Find the transmission and reflection coefficients. -(4)
Consider a particle incident from the left on the potential step. Where E = 2 eV...
Consider a particle incident from the left on the potential step. Where E = 2 eV V(x) {5 eV lo x < 0 x > 0 1) Find the wave function of the particle in two regions 2) Find reflection and transmission coefficients R and T
*Problem 2.27 Consider the double delta-function potential V (x) -α[6(x + a) + δ( )], x-...
*Problem 2.27 Consider the double delta-function potential V (x) -α[6(x + a) + δ( )], x- a where α and a are positive constants. (a) Sketch this potential. (b) How many bound states does it possess? Find the allowed energies, for a- h2/ma and for α h2/4ma, and sketch the wave functions.
6. (20pts) Consider a particle of mass m and energy E approaching the step potential V(x)...
6. (20pts) Consider a particle of mass m and energy E approaching the step potential V(x) = { 0x< V.>0 x > 0 from negative values of x. Consider the case E> Vo. a) Classically, what is the probability of reflection? b) Quantum mechanically, what is the probability of reflection? Express your result in terms of the ratio VIE. What is the probability of reflection if E= 2Vo?
toward a Problem 4 (30 points): Consider a current of particles of energy E moving from...
toward a Problem 4 (30 points): Consider a current of particles of energy E moving from x = - potential step as shown in the figure. x > 0 V(x) = {v. x<0 TE Where E > V a) (8 points) Derive the general solution of Schrödinger equation for x < 0 and for x > 0 b) (14 points) Apply the boundary conditions and calculate the transmission and the reflection coefficients. c) (8 points) What is the value of...
Figure 3. Double delta-function potential. X +a V(x) 2. Consider the symmetric, attractive double delta function potential illustrated in Fig. 3 where α is a positive constant. There are two lengths...
Figure 3. Double delta-function potential. X +a V(x) 2. Consider the symmetric, attractive double delta function potential illustrated in Fig. 3 where α is a positive constant. There are two lengths in this problem, the separation between the delta functions, 2a, and the decay lengthK-1-쁩)" of the wave function for an attractive delta function potential. [Note: In this problem, you may not need much math, but explain clearly the reasoning for your answers.] (a) How many bound states do you...
zone 1 Consider the following piecewise continuous, finite potential energy: ro; x < -a V(x)={-U, ;...
zone 1 Consider the following piecewise continuous, finite potential energy: ro; x < -a V(x)={-U, ; -a sxs a zone II U, > 0 (+ve) 10 ; x> a We consider zone III E>0: Unbound or scattering states (a) State the Time independent Schrödinger's Equation (TISE) and the expression of wave number k in each zone for the case of unbound state (b) Determine the expression of wave function u in each zone. (e) Determine the expression of probability Density...
Consider the finite rectangular barrier described by the potential: where Vo is a real and positive...
Consider the finite rectangular barrier described by the potential: where Vo is a real and positive constant. 1. How many bound states does this potential admit? 2. Find the transmission coefficient for a scattering state with energy E Vo 0
Consider a particle of a potential energy V(x) = {î where 1 is a positive constant....
Consider a particle of a potential energy V(x) = {î where 1 is a positive constant. ( V arises, for example from a force field - e.g a uniform electric field). Write Ehrenfest theorem for (h) and (p). Integrate these equations and compare the results to the classical results.
1. A particle is scattered upon by the potential V(x)-re(x). (β is a positive constant) (a) Write down the proper boundary conditions. (b) Find the reflection and transmission coefficients for E>0. (20%)
5. Consider a square potential barrier in figure below: V(x) 0 x <0 a) Assume that incident particles of energy E> v are coming from-X. Find the stationary states (the equations for region . 2 and 3 and the main equation for the all regions). Apply the matching limit conditions in the figure. Explain and find all the constants used in the equations in terms of the parameters provided and Planck's constant -(6) Find the transmission and reflection coefficients. -(4)
Consider a particle incident from the left on the potential step. Where E = 2 eV V(x) {5 eV lo x < 0 x > 0 1) Find the wave function of the particle in two regions 2) Find reflection and transmission coefficients R and T
*Problem 2.27 Consider the double delta-function potential V (x) -α[6(x + a) + δ( )], x- a where α and a are positive constants. (a) Sketch this potential. (b) How many bound states does it possess? Find the allowed energies, for a- h2/ma and for α h2/4ma, and sketch the wave functions.
6. (20pts) Consider a particle of mass m and energy E approaching the step potential V(x) = { 0x< V.>0 x > 0 from negative values of x. Consider the case E> Vo. a) Classically, what is the probability of reflection? b) Quantum mechanically, what is the probability of reflection? Express your result in terms of the ratio VIE. What is the probability of reflection if E= 2Vo?
toward a Problem 4 (30 points): Consider a current of particles of energy E moving from x = - potential step as shown in the figure. x > 0 V(x) = {v. x<0 TE Where E > V a) (8 points) Derive the general solution of Schrödinger equation for x < 0 and for x > 0 b) (14 points) Apply the boundary conditions and calculate the transmission and the reflection coefficients. c) (8 points) What is the value of...
Figure 3. Double delta-function potential. X +a V(x) 2. Consider the symmetric, attractive double delta function potential illustrated in Fig. 3 where α is a positive constant. There are two lengths in this problem, the separation between the delta functions, 2a, and the decay lengthK-1-쁩)" of the wave function for an attractive delta function potential. [Note: In this problem, you may not need much math, but explain clearly the reasoning for your answers.] (a) How many bound states do you...
zone 1 Consider the following piecewise continuous, finite potential energy: ro; x < -a V(x)={-U, ; -a sxs a zone II U, > 0 (+ve) 10 ; x> a We consider zone III E>0: Unbound or scattering states (a) State the Time independent Schrödinger's Equation (TISE) and the expression of wave number k in each zone for the case of unbound state (b) Determine the expression of wave function u in each zone. (e) Determine the expression of probability Density...
Consider the finite rectangular barrier described by the potential: where Vo is a real and positive constant. 1. How many bound states does this potential admit? 2. Find the transmission coefficient for a scattering state with energy E Vo 0
Consider a particle of a potential energy V(x) = {î where 1 is a positive constant. ( V arises, for example from a force field - e.g a uniform electric field). Write Ehrenfest theorem for (h) and (p). Integrate these equations and compare the results to the classical results.
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