a) Determine the energy of this particle, E. b) Show that the normalization constant, N, is given by - (W) ² N =
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a) Determine the energy of this particle, E.
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b) Show that the normalization constant, N, is given by shown in image
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- For a particle inside a box of finite potential well, the particle is most stable at what position of x? a) x > L b) x < 0 c) 0 < x < L d) Not stable in any stateSolve the following question completely mentioning each and every step3.) A classical ball bounces back and forth between two rigid walls with no loss of speed. After a long time has passed, the ball's position is measured at a random time. a. Is the probability Prob(inA)that the ball is found in sectionAlarger than, smaller than, or the same as the probability Prob(in B) of being found in section B?
- 1A particle in a 3-dimensional quadratic box with box length L has an energy given by h² E = (n+n+n). The degeneracies of the first, second, and 8mL² third level are a. e. 1, 2, 3 1, 3, 3 b. 1, 3, 1 c. 3, 3, 3 d. 1, 2, 2a) Write down the one-dimensional time-dependent Schro ̈dinger equation for a wavefunction Ψ(t, x) in a potential V (x). b) Write down the one-dimensional time-independent Schro ̈dinger equation for a wavefunc- tion ψ(x) in a potential V (x). c) Assuming that Ψ(t,x) corresponds to an energy eigenstate, write down a mathematical expression that relates the solutions of the one-dimensional time-dependent and time- independentSchro ̈dingerequations,Ψ(t,x)andψ(x).
- For a "particle in a box" of length, L, draw the first three wave functions and write down the wavelength of each. Confirm that the wavelengths for the nth level is given by 2L10. Ā = -2â + -3ŷ and B = -4â + -4ŷ. Calculate R = Ã+B. Calculate 0, the direction of R. Recall that 0 is defined as the angle with respect to the +x-axis. A. 49.4° B. 130.6° C. 229.4° D. 310.6°A particle trapped in a one-dimensional finite potential well with U0 = 0 in the region 0 < x < L, and finite U0 everywhere else, has a ground state wavenumber, k. The ground state wavenumber for the same particle in an infinite one-dimensional potential well of width L, would be equal to k. There is not enough information to determine. less than k. greater than k.
- Given 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)²7. Consider a particle in an infinite square well centered at x = 0 in one of its stationary states. For this problem, you may look up any integrals. Some useful ones are given in Harris. a) Compute (x) and (pr) for arbitrary n. Do this by direct computation but then describe how you could have found these results using symmetry (the symmetry can either be symmetry in the physical system, such as the shape of the wave function, or symmetry related to the expectation value integral, such as the shape of the integrand). b) Using your answer to part a), show that the uncertainty in the momentum is Apx nh for arbitrary n. Do this two ways: (i) first by using your answer to part a) and directly computating (p2) (via an integral) and (ii) by using your answer to part a) and relating (p2) to the kinetic energy operator. c) Show that the uncertainty principle holds for the ground state. 2L -A quantum mechanical particle of mass m moves in a 1D potential where a) Estimate the ground state energy of the particle. b) Sketch the wave function to the best of your ability.