Question

In: Physics

Asume the wave function Ψ(x) = A/(x²+a²) whith x real, A and a constants a) find...

Asume the wave function Ψ(x) = A/(x²+a²) whith x real, A and a constants

a) find the normalized wave function Φ(p) un the momentum space associated to Ψ(x)

b) use Φ(p) yo compute the expected values for p, p², and σ_p

c) verify if this state fulfills the Heisenberg uncertainty principle

Solutions

Expert Solution

All the parts have been attempted. The expectation values of <x> and <x^2>, though not asked in the question, are important to prove the uncertainty principle. Thus, they have been discussed in brief and the integrals involved can be solved very easily.


Related Solutions

A traveling wave along the x-axis is given by the following wave function ψ(x, t) =...
A traveling wave along the x-axis is given by the following wave function ψ(x, t) = 4.5 cos(2.1x - 11.8t + 0.52),where x in meter, t in seconds, and ψ in meters. Find a) the frequency, in hertz b)The wavelength in meters. c) The wave speed, in meters per second. d) The phase constant in radians.
A particle is described by the wave function ψ(x) = b(a2 - x2) for -a ≤...
A particle is described by the wave function ψ(x) = b(a2 - x2) for -a ≤ x ≤ a and ψ(x)=0 for x ≤ -a and x ≥ a, where a and b are positive real constants. (a) Using the normalization condition, find b in terms of a. (b) What is the probability to find the particle at x = 0.21a  in a small interval of width 0.01a? (c) What is the probability for the particle to be found between x...
A particle is described by the wave function ψ(x) = b(a2 - x2) for -a ≤...
A particle is described by the wave function ψ(x) = b(a2 - x2) for -a ≤ x ≤ a and ψ(x)=0 for x ≤ -a and x ≥ a , where a and b are positive real constants. (a) Using the normalization condition, find b in terms of a. (b) What is the probability to find the particle at x = 0.33a in a small interval of width 0.01a? (c) What is the probability for the particle to be found...
Consider the wave function Ψ = Ae−α|x| Which of the following boundary conditions are satisfied by...
Consider the wave function Ψ = Ae−α|x| Which of the following boundary conditions are satisfied by the wave function? Group of answer choices Ψ approaches zero as x approaches ±∞ Ψ is single valued. Ψ is finite everywhere. None of the boundary conditions are satisfied.
The parity operator P is defined by ˆ Pψ (x) =ψ (−x) for any function ψ(x)....
The parity operator P is defined by ˆ Pψ (x) =ψ (−x) for any function ψ(x). (a) Prove that this parity operator P is Hermitian. (b) Find its eigenvalues, and also its eigenfunctions (in terms of ψ(x)). (c) Prove that this parity operator commutes with the Hamiltonian when potential V(x) is an even function.
A particle's wave function is ψ(x) = Ae−c(x−b)2 where A = 1.95 nm−1/2 and b =...
A particle's wave function is ψ(x) = Ae−c(x−b)2 where A = 1.95 nm−1/2 and b = 0.600 nm. (a) What is the value of the constant c (in nm−2)? nm−2 (b) What is the expectation value for the position of this particle (in nm)? nm
The normalized wave function of an electron in a linear accelerator is ψ = (cos χ)eikx...
The normalized wave function of an electron in a linear accelerator is ψ = (cos χ)eikx + (sin χ)e–ikx, where χ (chi) is a parameter. (a) What is the probability that the electron will be found with a linear momentum (a) +kħ, (b) −kħ? (c) What form would the wave function have if it were 90% certain that the electron had linear momentum +kħ? (d) Evaluate the kinetic energy of the electron.
The wave function for hydrogen in the 1s state may be expressed as ψ(r) = Ae−r/a0....
The wave function for hydrogen in the 1s state may be expressed as ψ(r) = Ae−r/a0. Determine the normalization constant A for this wave function. (Use the following as necessary: a0.)
Partial differential equation (∂2Ψ/∂x2) – (∂2Ψ/∂y2 ) = 0                       Ψ = Ψ(x,y), i-Find the general...
Partial differential equation (∂2Ψ/∂x2) – (∂2Ψ/∂y2 ) = 0                       Ψ = Ψ(x,y), i-Find the general solution of this partial differential equation by using the separation of variables ii-Find the general solution of this partial differential equation by using the Fourier transform iii-Let Ψ(-L,y) = Ψ(L,y) , and Ψ(x,0) = Ψ(x,L) =0. Write the specific form of the solution you have found in either of part b).
Verify by direct substitution that the real functions Ψ = Acos (kx - wt) and Ψ...
Verify by direct substitution that the real functions Ψ = Acos (kx - wt) and Ψ = Asin (kx - wt) are not solutions of the SchroÈdinger equation for a free particle.vVerify by direct substitution that the real functions Ψ = Acos (kx - wt) and Ψ = Asin (kx - wt) are not solutions of the SchroÈdinger equation for a free particle.Verify by direct substitution that the real functions Ψ = Acos (kx - wt) and Ψ = Asin...
ADVERTISEMENT
ADVERTISEMENT
ADVERTISEMENT