### Quantum Mechanics: Particle in a Potential Field**Particle Description:**Consider a particle of mass \( m \) with energy \( E \), moving to the right from negative infinity. The particle is subject to a potential energy \( V(x) \) defined as:\[V(x) = \begin{cases} 0 & x \leq 0 \\V_0 & 0 \leq x < a \\\infty & x \geq a \end{cases}\]For all questions below, consider the case where \( E > V_0 \).**Tasks:**1. **Potential Energy Representation:** - Sketch a graph showing the potential energy \( V(x) \). This graph should represent a particle moving to the right when \( x < 0 \).2. **Time Independent Schrödinger Equation:** - Solve the time-independent Schrödinger equation to find the wave function \( \psi_E(x) \) for the domain \( -\infty < x \leq a \).3. **Probability of Reflection:** - Calculate the probability of reflection at the interface \( x = 0 \).4. **Wave Function Characteristics:** - Identify which part of the wave function represents a leftward-moving particle when \( x < 0 \). Demonstrate that this part of the wave function is an eigenfunction of the momentum operator, and calculate its eigenvalue.5. **Eigenfunction Verification:** - Determine if the wave function \( \psi_E(x) \) is an eigenfunction of the momentum operator for \( x < 0 \). Provide reasoning using words and/or mathematical proof for full credit.6. **Probability Current Density:** - Calculate the probability current density for the rightward-moving particle for \( x < 0 \). - Determine the total probability current density for the particle in the region where \( x < 0 \).**Additional Note:**- The probability current density \( J_x \) is given by: \[ J_x = \frac{\hbar}{2m} \left[ \psi^* \frac{\partial \psi}{\partial x} - \psi \frac{\partial \psi^*}{\partial x} \right] \]This exercise dives into quantum mechanics, focusing on concepts such as potential energy, wave functions, reflection probability, and

Note :- Since we only answer up to 3 sub-parts, we’ll answer the first 3. Please resubmit the…

Answered: Consider a particle of mass, m, with…

Similar questions

Please do fast and add explanation and check answer properly......... Consider two identical particles trapped in a one - dimensional box of size a . Write the symmetric and antisymmetric total wave functions considering that a particle is in the ground state and the other is in the first excited state, demonstrating this property. Also check that these wave functions are normalized.
a particle is confined to move on a circle's circumference (particle on a ring) such that its position can be described by the angle ϕ in the range of 0 to 2π. This system has wavefunctions in the form Ψm(ϕ)= eimlϕ where ml is an integer. Show that the wavefunctions Ψm(ϕ) with ml= +1 and +2 are ORTHOGONAL Show full and complete procedure. Do not skip any step
Part B and C please!
please solve number 4
The Hamiltonian for the one dimensional quantum oscillator is 1 p² 1 Ĥ = 1² + ½ k²² = 12 + √ mw² à² 2m 2m 2 where k = mw². 1) Define the operators ₁₁ and ₁₁ such that Ĥ = ½ħw (p² + ²). Define Ĥ2 as a function of 1 and p₁ such that Ĥ = hwĤ₂. - 2) Let us define the new operators â (1 + i₁) and â† = ½(î₁ — ip₁). Express ₁ and p₁ as a function of â and â³. Knowing that [^^1,î₁] = i and [1, 1] = -i, calculate âât and â†â. Express Ĥ2 as a function of a and at. 3) Let us define Ñ such that Ĥ₂ = Ñ + ½. Knowing that Ĥ, Ĥ₂ and Ñ have the same eigenstates, what are their corresponding eigenvalues?
The wavefunction for the motion of a particle on a ring is of the form ψ=NeimΦ . Evaluate the normalization constant, N. Show full and complete procedure in a clear way. DO NOT SKIP ANY STEP
The Hamiltonian of an electron of mass m in a constant electric field E in one dimension can be written as Ĥ=+eEx where â and are the position and momentum operators, respectively. With initials conditions (t = 0) = 0 and p(t = 0) = 0, which one of the following gives (t) at time in the Heisenberg picture? You may use the commutator [â,p] = iħ. O a. O b. eEt2 2m O C. e Et O d. -eEt O e. eEt² m pt m
Consider the motion of a point charge q in an electromagnetic field. Let Ē and 3 be the electric and magnetic fields, respectively, which can be derived from a scalar potential and a vector potential A ƏÃ Ē=-V6- - Ət Use the potential energy given by V(r) = 94 - 9(Ã· F). a. What is the lagrangian in terms of r? b. What is the conjugate momentum vector p. ? Show your solution. B = V X A
a) For a particle described by the wavefunction in Equation (1), show that the expectation values for momentum and momentum-squared are given by shown in image b) For the same particle, show that the expectation values for position and position-squared are given by shown in image