We will study the time-evolution of a finite dimensional quantum system. To this end, let us consider a quantum mechanical system with the Hilbert space&nbsp;C2. We denote by |0⟩ and |1⟩ the standard basis elements&nbsp;(1,0)T&nbsp;and&nbsp;(0,1)T. Let the Hamiltonian of the system in this basis be given byH=ω0110=0-i-i0and assume that for&nbsp;t=0&nbsp;the state of the system is just given by&nbsp;ψ(t=0)=∣0⟩. In the following, we also assume natural units in which&nbsp;h=―1.We expand the state at time t in the basis |0⟩, |1⟩ so:∣Psi(t)⟩=α0(t)∣0⟩+α1(t)∣1⟩Problems: Use Schrödinger's equation in order to derive a differential equations for&nbsp;α0&nbsp;and&nbsp;α1:(i) Find a solution given the initial conditions.(ii) What is the probability that the system can be measured in |1⟩ at some time t?

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We will study the time-evolution of a finite dimensional quantum system. To this end, let us consider a quantum mechanical system with the Hilbert space&amp;nbsp;C2. We denote by |0⟩ and |1⟩ the standard basis elements&amp;nbsp;(1,0)T&amp;nbsp;and&amp;nbsp;(0,1)T. Let the Hamiltonian of the system in this basis be given byH=ω0110=0-i-i0and assume that for&amp;nbsp;t=0&amp;nbsp;the state of the system is just given by&amp;nbsp;ψ(t=0)=∣0⟩. In the following, we also assume natural units in which&amp;nbsp;h=―1.We expand the state at time t in the basis |0⟩, |1⟩ so:∣Psi(t)⟩=α0(t)∣0⟩+α1(t)∣1⟩Problems: Use Schrödinger&#039;s equation in order to derive a differential equations for&amp;nbsp;α0&amp;nbsp;and&amp;nbsp;α1:(i) Find a solution given the initial conditions.(ii) What is the probability that the system can be measured in |1⟩ at some time t?

mislifola5vo · Accepted Answer

Step 1|Ψ(t)&amp;gt;&amp;gt;=(α0(t)α1(t))Step 2Then the Schrödinger equation tells us:ih∂∂t―∣Psi(t)⟩=H∣Psi(t)⟩iℏ∂∂tα0(t)α1(t)=0-i-i0α0(t)α1(t)Although I will say I&#039;m a little confused as to why the Hamiltonian isn&#039;t diagonalized in what I&#039;m assuming is the energy basis given this a two level system.

We will study the time-evolution of a finite dimensional quantum system. To this end, let us conside

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