. Evaluate the ground state energy of a harmonic oscillator of mass 'm' and angular frequency o'using the trial function. $(x) = { cos (2a) -a sxsa

Q: 1. A particle of m moves in the attractive central potential: V(r) = ax6, where a is a constant and…

A: Ans 1:  (a) A=(π2b​)1/4. (b) E(b)=2mbℏ2​+64b315α​. (c) bmin​=(32ℏ245αm​)1/4. (d)…

Q: A harmonic oscillator is in the state o (3) = N (3² – 3y + 1) exp (-) (a) Expressed the state…

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Q: Consider the 1-dimensional quantum harmonic oscillator of mass u. xeax*/½ is an eigenfunction of the…

A: Given: 1 dimensional quantum harmonic oscillator.  We have to show that ψ = xeαx2/2 is eigen…

Q: Use the ground-state wave function of the simple har- monic oscillator to find x, (x²), and Ax. Use…

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Q: Find the angular momentum and kinetic energy in the z axis for the cos30e®+ sin30e-º wave function.

A: Given that,ψ=Cos 30 eiϕ+Sin 30 e-iϕwe know that,Lz=-ih∂∂ϕK.Ez=p22mz=-h22m1r2 sin2θ∂2∂ϕ2hence,…

Q: A particle of mass 10 kg is attached to a spring of spring constant 10³ N/m. Calculate its zero…

A: The zero-point energy of a harmonic oscillator is given by the equation:

Q: The Longrongion of 1D harmonic oscilator is, () write Euler-Congrange equation of system? ().write…

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Q: What is the first excited state energy for a square well potential (with V = -10 hartrees and a…

A: Given, V= -10  hartrees width of -1 < x < 1

Q: Consider a particle in the one-dimensional box with the following wave function: psi(x, 0) = Cx(a−x)

A: Given a particle in a 1-D box having a wave function ψx,0=Cx(a-x) We need to find  dx^dtanddp^dt…

Q: Calculate the average potential energy of the second excited state of the oscillator. harmonic given…

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Q: Obtain the ground state of 1D harmonic oscillator using variational method

A: Variational method is used in quantum mechanics to make an approximate idea of lowest energy states.…

Q: (2x – ) is a suitable wavefunction for a 1-dimensional particle-in-a-box where the box Y = cos…

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Q: 1. Z{2cos[0.2(n – 1)] u(n)} -

A: Given: Z{2 cos [0.2π(n-1)]u(n)}

Q: A particle of mass m, which moves freely inside an infinite potential well of length a, is initially…

A: Given: ψ(x,0)=35asin3πxa+15a sin5πxa For the wavefunction at a later time t we have;…

Q: ITX Consider the wavefunction y(x) = D sin exp(i Tx ) for 0<X <L, find: L a) The constant D b) The…

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Q: Find the normalize constant A and the average value of the kinetic energy of a particle in box has…

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Q: At time t = 0, a rigid rotor is in a state whose functional form in configuration space can be…

A: Given: The wavefunction of the rigid rotor is

Q: > show that the time independ ent schrodinger equation for a partide teapped in a 30 harmonic well…

A: Solution attached in the photo

Q: The wavefunction for a harmonic oscillator with v= 1 is given by: Vv-1 Ny 2y exp 2 Determine the…

A: I have use concept of mean value

Q: The wavefunction of the n-th energy level of a harmonic oscillator can be expressed as: with: Vn(x)…

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Q: How would you numerically calculate the ground state of a displaced harmonic oscillator H woâtà +…

A: The displaced harmonic oscillator is given as  H^=ω0a^+a^+λω0a^++a^ where a^+ is creation operator…

Q: 1. Let nem (r) = Rne(r)Y(0, 0) be the solutions to the stationary Schroedinger equation with a…

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Q: B) Suppose an harmonic oscillator in state (1) Calculate the expectation value of x?

A: Given that The state of a harmonic oscillator is |1&gt;   And we have to find the expectation value…

Q: A one-dimensional simple lharmonic oscillator is subjected to a small perturbing potential &V(2),…

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Q: Which of the wave functions.below represents a solution mathematically Schodinger equation 1:) V (r,…

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Q: A free particle has the initial wave function (.x, 0) = Ae-a.x² where A and a are constants (a is…

A: Given Initial wave function ψ x,0 = Ae-ax2 Where, A and a = constant

Q: Find the angular momentum and kinetic energy in the z axis for the Cos30eiΦ+ Sin30e-iΦ wave…

A: given that, ψ = cos30 eiϕ + sin30 e-iφ we need to find, angular momentum Lz  and kinetic energy K.E…

Q: Apply variational method to simple harmonic oscillator . Use different trial wavefunctions and…

A: Taking an exponentially decreasing trail wavefunction: ψ(x)=Ae-βx

Q: -state syste Fgy 0, e, 2e, monic oscil

A: a)  Partition function of n state : Z = ∑k=0n  e-βk∈    = ∑k=0ne-β∈kZ =1 -e-β∈(n+1)1 -e-β∈   --(1)…

Q: A partick of mass m in the potential (x, y)= (4x +²), is in an eigenstate of enagy *: The…

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Q: -h? d? 3. Find average value of kinetic energy, for ground state of the harmonic 2µ dx? ocillator.…

A: We have to use some basic formula here

Q: 1. A particle of m moves in the attractive central potential: V(r) = ax6, where a is a constant and…

A: The objective of the question is to compute the normalization constant A, calculate the ground state…

Q: 7. (a) A one dimensional harmonic oscillator in its ground state is perturbed by V(x, t) Ar sin² wt.…

A: One harmonic oscilator,