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eE7B2(a) Normalize (to 1) the wavefunction e-ax2 ǐn the range-oo a >0. Refer to the Resource...
17. If F is a d.f. such that F(0-) = 0, then -F)) dx xdF(x) +oo...
17. If F is a d.f. such that F(0-) = 0, then -F)) dx xdF(x) +oo Thus if X is a positive r.., then we have P[X x}dx E(X) PX> x}dx =
PROBLEMS 2.32 The wavefunction for a particle in one dimension is given by Another state that...
PROBLEMS 2.32 The wavefunction for a particle in one dimension is given by Another state that the particle may be in is A third state the particle may be in is y2/4 Normalize all three states in the interval-oo < y <-co (i.e., find A1,A2, and A3) is the probability of finding the particle in the interval 0 y < 1 when the particle : is in the state vs the same as the sum of the separate probabilities for...
(C) An electron is described by the wavefunction (x) = 4 cos(2x/L) for the range =...
(C) An electron is described by the wavefunction (x) = 4 cos(2x/L) for the range = 5234 and is zero otherwise. (In other words, v(x) = 0 for 3 and 43 .) A useful integral is S cos? (ax)dx = 1 + sin (2017) (1) What is the probability of finding the electron between x = 0 and x = ? (ii) What is the probability of finding the electron at = 4? (iii) Where is the maximum probability for...
Please solve the normalization, 7e, and the commutator questions 6. Normalization Normalize the following functions: sin...
Please solve the normalization, 7e, and the commutator
questions
6. Normalization Normalize the following functions: sin (1") between 0<x<L 200 for 0 <r <o, treating do as a constant 7. Eigenfunctions and Eigenvalues Determine which of the following are eigenfucntions of the operator 4 give the eigenfunction. Where appropriate (a) pikx (b) cos ka (c) k (d) kx (e) e-ax? 8. Commutator Evaluate the commutator (î, P2]
2. Let f(x,y) = e-r-u, 0 < x < oo, 0 < y < oo, zero...
2. Let f(x,y) = e-r-u, 0 < x < oo, 0 < y < oo, zero elsewhere, be the pdf of X and Y. Then if Z = X + Y, compute (a) P(Z 0). (b) P(Z 6) (c) P(Z 2) (d) What is the pdf of Z?
3. Let X be a continuous random variable with probability density function ax2 + bx f(0)...
3. Let X be a continuous random variable with probability density function ax2 + bx f(0) = -{ { for 0 < x <1 otherwise 0 where a and b are constants. If E(X) = 0.75, find a, b, and Var(X). 4. Show that an exponential random variable is memoryless. That is, if X is exponential with parameter > 0, then P(X > s+t | X > s) = P(X > t) for s,t> 0 Hint: see example 5.1 in...
,X, ,n. independent, the central Xi, E(X)=0, var(X)-σ are Prove 3. Assume <o。 13<oo, 1=1, limit...
,X, ,n. independent, the central Xi, E(X)=0, var(X)-σ are Prove 3. Assume <o。 13<oo, 1=1, limit theorem (CLT) based EX1 result regarding what are conditions on σ that we need to assume in order for the x.B.= Σσ, as n →oo. In this context, X,, B" =y as n →oo, In this context, result to hold?
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)
A little blurry, but the wavefunction is a^(1/2)*e^-(ax/2). Not sure how to find expectation value of...
A little blurry, but the wavefunction is a^(1/2)*e^-(ax/2). Not
sure how to find expectation value of the commutator. (What is the
commutator of this wavefunction?)
Ppie P7C. A particle is in a state described by the normalized wavefunction vex) ewhere a is a constant and 0 s xS oo, Evaluate the expectation value of the commutator of the position and momentum operators.
(12) Suppose that f [0, oo) - [0, o0) and that f E R(0, n), for...
(12) Suppose that f [0, oo) - [0, o0) and that f E R(0, n), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral So.o)dexists, and f dA f (x)dax lim -- noo 0,00)
17. If F is a d.f. such that F(0-) = 0, then -F)) dx xdF(x) +oo Thus if X is a positive r.., then we have P[X x}dx E(X) PX> x}dx =
PROBLEMS 2.32 The wavefunction for a particle in one dimension is given by Another state that the particle may be in is A third state the particle may be in is y2/4 Normalize all three states in the interval-oo < y <-co (i.e., find A1,A2, and A3) is the probability of finding the particle in the interval 0 y < 1 when the particle : is in the state vs the same as the sum of the separate probabilities for...
(C) An electron is described by the wavefunction (x) = 4 cos(2x/L) for the range = 5234 and is zero otherwise. (In other words, v(x) = 0 for 3 and 43 .) A useful integral is S cos? (ax)dx = 1 + sin (2017) (1) What is the probability of finding the electron between x = 0 and x = ? (ii) What is the probability of finding the electron at = 4? (iii) Where is the maximum probability for...
Please solve the normalization, 7e, and the commutator
questions
6. Normalization Normalize the following functions: sin (1") between 0<x<L 200 for 0 <r <o, treating do as a constant 7. Eigenfunctions and Eigenvalues Determine which of the following are eigenfucntions of the operator 4 give the eigenfunction. Where appropriate (a) pikx (b) cos ka (c) k (d) kx (e) e-ax? 8. Commutator Evaluate the commutator (î, P2]
2. Let f(x,y) = e-r-u, 0 < x < oo, 0 < y < oo, zero elsewhere, be the pdf of X and Y. Then if Z = X + Y, compute (a) P(Z 0). (b) P(Z 6) (c) P(Z 2) (d) What is the pdf of Z?
3. Let X be a continuous random variable with probability density function ax2 + bx f(0) = -{ { for 0 < x <1 otherwise 0 where a and b are constants. If E(X) = 0.75, find a, b, and Var(X). 4. Show that an exponential random variable is memoryless. That is, if X is exponential with parameter > 0, then P(X > s+t | X > s) = P(X > t) for s,t> 0 Hint: see example 5.1 in...
,X, ,n. independent, the central Xi, E(X)=0, var(X)-σ are Prove 3. Assume <o。 13<oo, 1=1, limit theorem (CLT) based EX1 result regarding what are conditions on σ that we need to assume in order for the x.B.= Σσ, as n →oo. In this context, X,, B" =y as n →oo, In this context, result to hold?
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)
A little blurry, but the wavefunction is a^(1/2)*e^-(ax/2). Not
sure how to find expectation value of the commutator. (What is the
commutator of this wavefunction?)
Ppie P7C. A particle is in a state described by the normalized wavefunction vex) ewhere a is a constant and 0 s xS oo, Evaluate the expectation value of the commutator of the position and momentum operators.
(12) Suppose that f [0, oo) - [0, o0) and that f E R(0, n), for every n E N. Prove that f is Lebesgue measurable, the Lebesgue integral So.o)dexists, and f dA f (x)dax lim -- noo 0,00)
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