Suppose that the probability of observing |0⟩ in the state |ϕ1⟩ is 1/4 and the probability of observing |1⟩ in the qubit |ϕ2⟩ is 1/3. Find the probability of observing |10⟩ in the state |ϕ1ϕ2⟩.
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Suppose that the probability of observing |0⟩ in the state |ϕ1⟩ is 1/4 and the probability of
observing |1⟩ in the qubit |ϕ2⟩ is 1/3.
Find the probability of observing |10⟩ in the state |ϕ1ϕ2⟩.
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- Let n (x) denote the orthonormal stationary states of a system corresponding to the energy En. Suppose that the normalized wave function of the system at time t = 0 is þ(x,0) and suppose that a measurement of the energy yields the value E1 with probability 1/2, E2 with probability 3/8, and E3 with probability 1/8. (a) Write the most general expansion for Þ(x,0) consistent with this information. (b) What is the expansion for the wave function of the system at time t, Þ(x, t)?The coherent states for the one-dimensional harmonic oscillator are defined as eigenstates of the operatorof annihilation a (which is non-Hermitian):a |λ⟩ = λ |λ⟩ (1)where λ is a complex number in general. a)prove that is a normalized consistent state. b)Show that the above state satisfies the minimum uncertainty relation, i.e., show thatSuppose that a qubit has a state of the form |ϕ⟩ = α |0⟩ + β |1⟩. If the probability of measuring the value |1⟩ in the qubit is |2/3i-4| then find the probability of measuring the the value |0⟩ in the qubit
- Show that at high enough temperatures (where KBT » ħw) the partition function of a simple quantum mechanical harmonic oscillator is approximately Z≈ (Bħw)-¹ Then use the partition function to calculate the high temperature expressions for the internal energy U, the heat capacity Cy, the Helmholtz function F and the entropy S.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).For a system of fermions at room temperature, compute the probability of a single-particle state being occupied if its energy is. 0.01 eV less than μ
- 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.Determine the expectation values of the position (x) (p) and the momentum 4 ħ (x)= cos cot,(p): 5V2mw 4 mah 5V 2 sin cot 2 ħ moon (x)= sin cot, (p)= COS at 52mo 2 4 h 4 moh (x)= 52mo sin cot.(p) COS 2 h s cot, (p) 5V2mco 2 moh 5V 2 sin of as a function of time for a harmonic oscillator with its initial state ())))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.
- An atom is in an excited state for 4.00 us before moving back to the ground state. Find the approximate uncertainty in energy of the photon in units of 10¹¹ eV. (A) 8.23 (B) 3.78 (C) 4.97 (D) 5.49 (E) 6.17Consider a one-dimensional particle which is confined within the region 0≤x≤a and whose wave function is (x, t) = sin (x/a) exp(-iwt). (D) v sv (a) Find the potential V(x). (b) Calculate the probability of finding the particle in the interval a/4 ≤x≤3a/4.