Linear operators play an important role in the quantum mechanical description of matter. Which of the following is true of all linear operators, A and B? A[k f(x)]=k Ã(x) A[Ã(x)]=Ã(x) +à (x) A[Bƒ(x)]=B[Ã(x)] à ƒ(x) =ƒ(x)
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We will answer this question by looking at definition of linear operator.
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- 5. Consider a potential barrier represented as follows: U M 0 +a U(x) = x a Determine the transmission coefficient as a function of particle energy.No Spacing Heading 1 Normal Aa v A A 困、 Paragraph Styles The action along a path is defined to be: S = [(K.E.-P. E.) dt Determine the physical units of action. Detail Feynman's approach to calculating the probability amplitude for an electron to go from one event A to another B using the "sum over all paths". A'Focus 12 1> 12What’s Orthogonal Set in quantum mechanics? And Write its mathematical formula
- = = An electron having total energy E 4.60 eV approaches a rectangular energy barrier with U■5.10 eV and L-950 pm as shown in the figure below. Classically, the electron cannot pass through the barrier because E < U. Quantum-mechanically, however, the probability of tunneling is not zero. Energy E U 0 i (a) Calculate this probability, which is the transmission coefficient. (Use 9.11 x 10-31 kg for the mass of an electron, 1.055 x 10-34] s for h, and note that there are 1.60 x 10-19 J per eV.) (b) To what value would the width L of the potential barrier have to be increased for the chance of an incident 4.60-eV electron tunneling through the barrier to be one in one million? nmn=2 35 L FIGURE 1.0 1. FIGURE 1.0 shows a particle of mass m moves in x-axis with the following potential: V(x) = { 0, for 0A particle is described by the wavefunction Ψ(t, x), and the momentum operator is denoted by pˆ. a) Write down an expression for the differential operator pˆ. b) Write down an expression for the expectation value of the momentum, ⟨p⟩. c) Write down an expression for the probability density, ρ. d) Write down an expression for the probability of finding the particle between x = a and x = b.A) Report your answer as a decimal number with three signficant figures. B)Give your answer as a decimal number with three significant figures. C) How does the classical kinetic energy of the free electron compare in magnitude with the result you obtained in the previous part?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.Problem 3. Consider the two example systems from quantum mechanics. First, for a particle in a box of length 1 we have the equation h² d²v 2m dx² EV, with boundary conditions (0) = 0 and (1) = 0. Second, the Quantum Harmonic Oscillator (QHO) V = EV h² d² 2m da² +ka²) 1 +kx² 2 (a) Write down the states for both systems. What are their similarities and differences? (b) Write down the energy eigenvalues for both systems. What are their similarities and differences? (c) Plot the first three states of the QHO along with the potential for the system. (d) Explain why you can observe a particle outside of the "classically allowed region". Hint: you can use any state and compute an integral to determine a probability of a particle being in a given region.