25. Consider a particle of mass m in a one-dimensional infinite square well with V (x) = 0 for 0 ≤ x ≤ a and V(x) = ∞ elsewhere. A time-dependent perturbation is added of the form V₁(x, t) = Ɛ ( Xx 1 - e for 0 < x
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- 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).A particle with mass m is in the state .2 mx +iat 2h Y(x,t) = Ae where A and a are positive real constants. Calculate the expectation values of (x).1. (1) Derive one dimensional time-independent Schrödinger equation from the classical wave equation and P=h/h and (2) deduce momentum operator and kinetic energy operator for the wave function y. 2. Calculate the specific density function of a single particle, |v|2, on one-dimensional axis (x-axis) including the boundary condition as below; = 00 at x L 3. If the single particle of problem 2 is electron and the L is 10 Å, what are its momentum and energy of the electron with the quantum number of 2? And if there are 1 mole of electrons with the same environment, describe what is the total energy. (mass of electron =9.1 × 10 31 kg, Planck's constant, h= 6.63 × 10-3ª m² kg / s, and Avogadro's number is 6.02 x 10²*. You can choose any unit to show the answer.) 4. Dervie three dimensional time-independent Schrödinger equation for one-electron based on spherical coordinate from Cartesian coordinate system as below. Hemiltonian operator for one-electron system: -h? ( a2 H = 8n2m ax2 ay2 ™…
- 322. Using the normalization condition, show that the constant A has the value (ma/ħn)/4 for the one- dimensional simple harmonic oscillator in its ground state.A 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.
- Question A2 Consider an infinite square well of width L, with V = 0 in the region -L/2 < x < L/2 and V → ∞ everywhere else. For this system: a) Write down and solve the time-independent Schrödinger equation for & inside the well, where -L/2< xi need the answer quickly2. Consider a particle of mass m in an infinite square well, 0(≤ x ≤a). At the time t = 0 the particle is in the ground (n = 1). Then at t> 0 a weak time-dependent external potential is turned on: H' = Axe T