(a) Determine the value of C is the wave-function is normalised. (b) Where is the electron most likely to be found? (c) Calculate the expectation value of position < x > for the electron. (d) What is the uncertainty in finding the position of the electron?
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- WHY DOES THE WAVE- FUNCTION GO TO ZERO AS X GOES TO * INFINITY ?The wave function of a particle in a one-dimensional box of width L is u(x) = A sin (7x/L). If we know the particle must be somewhere in the box, what must be the value of A?An electron has a kinetic energy of 13.3 eV. The electron is incident upon a rectangular barrier of height 21.5 eV and width 1.00 nm. If the electron absorbed all the energy of a photon of green light (with wavelength 546 nm) at the instant it reached the barrier, by what factor would the electron's probability of tunneling through the barrier increase?
- The wave function of a particle at t = 0 is given as: ψ(x, t) = C exp[ -|x|/x0] where C and x0 are constants. (a) What is the relation between C and x0?(b) Calculate the expectation value of position x of the particle.(c) Suggest a region in x in which the probability of finding the particle is 0.5.(5) The wave function for a particle is given by: (x) = Ae-=/L for r 2 0, where A and L are constants, and L > 0. b(x) = 0 for r < 0. (a) Find the value of the constant A, as a function of L. A useful integral is: fe-K=dx = -ke-K, %3D where K is a constant. (b) What is the probability of finding the particle in the range –10 L < x< -L? (c) What is the probability of finding the particle in the range 0Given a Gaussian wave function: Y(x) = (1/a)-1/4e-ax²/2 Where a is a positive constant 1) Find the normalization (if the wave function is not normalized) 2) Determine the mean value of the position x of the particle : x 3) Determine the mean value of x? : x? 4) Determine the value of Ax = /(x²) – (x)²The 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, ....At time t = 0 the normalized wave function for a particle of mass m in the one-dimensional infinite well (see first image) is given by the function in the second image. Find ψ(x, t). What is the probability that a measurement of the energy at time t will yield the result ħ2π2/2mL2? Find <E> for the particle at time t. (Hint: <E> can be obtained by inspection, without an integral)In the lab you make a simple harmonic oscillator with a 0.15-kg mass attached to a 12-N/m spring. (a) If the oscillation amplitude is 0.10 m, what is the corresponding quantum number n for the quantum harmonic oscillator? (b) What would be the amplitude of the quantum ground state for this oscillator? (c) What is the energy of a photon emitted when this oscillator makes a transition between adjacent energy levels? Comment on each of your results.