A particle has the time-independent wave function Aebz r<0 v(x) = Ae¬bz x > 0 where b is known but arbitrary. Calculate the value of A that normalizes the probability associated with the wave function.
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- The particle is confined to a one-dimensional box between x=0 and x=2. Its wave function is A(x)=6x2/N (0≤x≤1) and A(x)=6/N (1<x≤2), where N is a normalization constant. Calculate the constant N and the average position of the particles.Show that at high enough temperatures (where KBT » ħw) the partition function of a simple quantum mechanical harmonic oscillator is approximately Z≈ (Bħw)-¹ Then use the partition function to calculate the high temperature expressions for the internal energy U, the heat capacity Cy, the Helmholtz function F and the entropy S.6QM Please answer question throughly and detailed.
- Consider the wavefunction for a particle in a one-dimensional box when the level is n = 6. Calculate the total probability of finding the particle between x = 0 and x = L/12? Provide your answer to three significant figures.Suppose we had a classical particle in a frictionless box, bouncing back and forth at constant speed. The probability density of the position of the particle in soma box of length L is given by: 0 ans-fawr (7) p(x)= 0 x L a. Sketch the probability density as a function of position b. What must A be in order for p(x) to be normalized? Remember that you are welcome to use resources to solve integrals such as Wolfram Alpha, a table of integrals etc.A particle with mass m is in the state mx +iat 2h V (x, t) = Ae where A and a are positive real constants. Calculate the expectation value of (p).
- The wave function W(x,t)=Ax^4 where A is a constant. If the particle in the box W is normalized. W(x)=Ax^4 (A x squared), for 0<=x<=1, and W(x) = 0 anywhere. A is a constant. Calculate the probability of getting a particle for the range x1 = 0 to x2 = 1/3 a. 1 × 10^-5 b. 2 × 10^-5 c. 3 × 10^-5 d. 4 × 10^-5Consider the wavefunction Y(x) = exp(-2a|x|). a) Normalize the above wavefunction. b) Sketch the probability density of the above wavefunction. c) What is the probability of finding the particle in the range 0 < x s 1/a ?The expectation value of a function f(x), denoted by (f(x)), is given by (f(x)) = f(x)\(x)|³dx +00 Yn(x) = where (x) is the normalised wave function. A one-dimensional box is on the x-axis in the region of 0 ≤ x ≤ L. The normalised wave functions for a particle in the box are given by -sin -8 Calculate (x) and (x²) for a particle in the nth state. n = 1, 2, 3, ....