9. A particle of mass m is in the state (x, t) = Ae¯ª[(mx²/ħ)+it] where A and a are positive real constants. (a) Find the normalization constant A. (b) For what potential energy function U(x) does Y satisfy the Schrodinger equation (c) Calculate and
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- Solve the time-independent Schrödinger equation and determine the energy levels and the wave function of a particle in the potential a? V (x) = Vol a + 2r2 with a = const.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.QUESTION 7 Use the Schrödinger equation to calculate the energy of a 1-dimensional particle-in-a-box system in which the normalized wave function is 4' = e sin(6x). The box boundaries are at x=0 and x=r/3. The potential energy is zero when 0 < x <- and o outside of these boundaries. 18h? m h2 8m h2 36n2m none are correct
- Which of the following is/are correct for the equation y(x) dx defined for a particle whose state function is y(x) (11) (iii) This equation gives the probability of the particle with the range x to X₂. This equation applies to the particle moving in any dimension. This equation defines relation between the state function and the probability with the range x; to x₂- (a) Only (1) (b) (ii) and (iii) (c) (i) and (iii) (d) (i) and (ii)4) Consider the one-dimensional wave function given below. (a) Draw a graph of the wave function for the region defined. (b) Determine the value of the normalization constant. (c) What is the probability of finding the particle between x = o and x = a? (d) Show that the wave function is a solution of the non-relativistic wave equation (Schrodinger equation) for a constant potential. (e) What is the energy of the wave function? (x) = A exp(-x/a) for x > o (x) = A exp(+x/a) for x < oA qubit is in state |) = o|0) +₁|1) at time t = 0. It then evolves according to the Schrödinger equation with the Hamiltonian Ĥ defined by its action on the basis vectors: Ĥ0) = 0|0) and Ĥ|1) = E|1), where E is a constant with units of energy. a) Solve for the state of the qubit at time t. b) Find the probability to observe the qubit in state 0 at time t. Explain the result by referring to the way that the time-evolution transforms the Bloch sphere.