We have potential of V (x) = ( 0, 0 ≤ x ≤ a. ∞, elsewhere....

We have potential of

V (x) = ( 0, 0 ≤ x ≤ a.
∞, elsewhere.

a) Find the ground state energy and the first and second excited states, if an electron is enclosed in this potential of size a = 0.100 nm.

b) Find the ground state energy and the first and second excited states, if a 1 g metal sphere is enclosed in this potential of size a = 10.0 cm.

c) Are the quantum effects important for both systems? Explain why or why not.

d) Use the uncertainty principle to estimate the velocities of the electron and the metal sphere.

Expert Solution

genius_generous answered 2 years ago

Bound state in potential. Where V(x)=inf for x<0 V(x)=0 for 0 ≤ x ≤ L V...

Bound state in potential. Where V(x)=inf for x<0 V(x)=0 for 0 ≤ x ≤ L V (x) = U for x > L Write down the Schrödingerequation and solutions (wavesolutions) on general form for the three V(x).

Consider the “step” potential: V(x) = 0 if x ≤ 0 = V0 if...

Consider the “step” potential: V(x) = 0 if x ≤ 0 = V0 if x ≥ 0 (a) Calculate the reflection coefficient for the case E < V0 , and (b) for the case E > V0.

Consider the step potential V (x) = 0 x ≤ 0 (I) = V0 x >...

Consider the step potential V (x) = 0 x ≤ 0 (I) = V0 x > 0 (II) . (a) What does the wave function for the scattering problem with 0 < E < V0 look like in regions I and II? (Write the equation for the wave functions, including only terms with non-zero coeﬃcients.) (b) What are the boundary conditions for the wave function and its derivative at x = 0? Deﬁne constants you are using (e.g., κ, k,...

The potential in a region between x = 0 and x = 6.00 m is V...

The potential in a region between x = 0 and x = 6.00 m is V = a + bx, where a = 12.6 V and b = -7.90 V/m. (a) Determine the potential at x = 0. ______ V Determine the potential at x = 3.00 m. ______ V Determine the potential at x = 6.00 m. ______ V (b) Determine the magnitude and direction of the electric field at x = 0. magnitude ________ V/m direction +x or...

Consider the delta function potential well :V(x) = −αδ(x) (α > 0). (a) What are the...

Consider the delta function potential well :V(x) = −αδ(x) (α > 0). (a) What are the boundary conditions on the bound state wave function and its derivative at x = ±∞ and x = 0? Explain. Hint: You may find the property stated in Prob. 2 useful. (b) Show that there is only one bound state. (c) Find the energy E and wave function ψ(x) of the bound state. (d) Find the transmission coefficient for scattering states, with energy E...

(a) Let X and Y have the joint pdf ???(?, ?)=1, 0≤x≤3/2, 0≤y≤1, zero elsewhere. Find:...

(a) Let X and Y have the joint pdf ???(?, ?)=1, 0≤x≤3/2, 0≤y≤1, zero elsewhere. Find: 1 The pdf of Z=X+Y 2 The pdf of Z=X.Y

Let X be an exponential distribution with mean=1, i.e. f(x)=e^-x for 0<X< ∞, and 0 elsewhere....

Let X be an exponential distribution with mean=1, i.e. f(x)=e^-x for 0<X< ∞, and 0 elsewhere. Find the density function and cdf of a) X^1/2 b)X=e^x c)X=1/X Which of the random variables-X, X^1/2, e^x, 1/X does not have a finite mean?

Let f(x) = a(e-2x – e-6x), for x ≥ 0, and f(x)=0 elsewhere. a) Find a...

Let f(x) = a(e-2x – e-6x), for x ≥ 0, and f(x)=0 elsewhere. a) Find a so that f(x) is a probability density function b)What is P(X<=1)

The potential of a particle is given as V (x) = -Vδ (x), with V being...

The potential of a particle is given as V (x) = -Vδ (x), with V being a positive number. Find the wave function and energy of the bounded states of this particle.

Let f(x) = {(C/x^n if 1≤ x <∞; 0 elsewhere)} where n is an integer >1....

Let f(x) = {(C/x^n if 1≤ x <∞; 0 elsewhere)} where n is an integer >1. a. Find the value of the constant C (in terms of n) that makes this a probability density function. b. For what values of n does the expected value E(X) exist? Why? c. For what values of n does the variance var(X) exist? Why?

Question