Functions of the form cos(nπx/L), where n = 1, 3, 5 …, can be used to model the wavefunctions of particles confined to the region between x = −L/2 and x = +L/2. The integration is limited to the range −L/2 to +L/2 because the wavefunction is zero outside this range. Show that the wavefunctions are orthogonal for n = 1 and 3. You will find the necessary integral in the Resource section.

Q: Find the energy of plane wave function exp i (kx-wt)

A: Given, Wave function, ψ=eikx-wt

Q: For the particle in a box, we chose k = np/L with n = 1, 2, 3, c to fit the boundary condition that…

A:

Q: dimensional wave packet at time t=0 (k)=0 for k

A: The wave function is given as:

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

A:

Q: by y Cxst) fotential Vcx wave functian deserbing aParticle is qixen - Sin () -i wt 7Xx4 fined the…

A:

Q: If p=-iħ d/dx is the linear momentum operator. What is its eigenvalue if it is affected on a…

A: if any operator operates on wave function, than the result should be  some constant multiplied by…

Q: (c) Let the wave function for the particle is (r) e Prove it is eigenstate of the kinetic energy %3D

A: Given data, Wave function of the particle : ψx=12eikxx

Q: For the infinite square-well potential, fi nd the probability that a particle in its ground state is…

A: Given information: An infinite square-well potential. We have to find the probability that a…

Q: Consider the first excited state of the quantum harmonic oscillator (v = 1) and the wavefunction…

A: For quantum harmonic oscillator, its position extends from..For a wave function  to be normalized,…

Q: Calculate the uncertainties dr = V(r2) and dp = Vp?) for a particle confined in the region -a a, r…

A: As we can see the given wave function is normalised and in outside region it's zero. Therefore This…

Q: As a 1-dimensional problem, you are given a particle of mass, m, confined to a box of width, L. The…

A:

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

A:

Q: The wavefunction for the particle in a one-dimensional infinite potential well is given by n2n?h?…

A: In part 1 use the normalisation condition and prove that the given wavefunction is normalised, In…

Q: H.W.: prove that the normalization constant for the wavefunction , (x) = A sin(-x) %3D 2a Equal to…

A: Given data, Wave function is given as :- ψnx=Asinnπx2a    which is the wave function of a particle…

Q: Show that the hydrogenic wavefunctions y1, and y2, are normalized. mutually orthogonal and

A:

Q: Following is a 1D wavefunction that is associated with a particle moving between o and +oo: (x) =…

A: We will use basic principles of QM to solve

Q: Normalize the wavefunction re-r/2a in three-dimensional space

A: Solution  The wavefunction  re-r/2a in the three dimensional space is given by

Q: the wave function everywhere.

A:

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

A:

Q: Don't use chat gpt It

A:

Q: if the particle has the wavelenght si =a*x^2 between x = 0 and si =0 elsewhere. find the probability…

A:

Q: For the scaled stationary Schrödinger equation "(x) + 8(x)v(x) = Ev(x), find the eigenvalue E and…

A:

Q: A particle with the energy E is incident from the left on a potential step of height Uo and a…

A: This problem is a combination of step and delta function. There are two regions here, one is x<0,…

Q: im@ Find the expectation Value (L₂) of the wave function 10-e' and prove the ² Where Lz - ih == Ә…

A:

Q: (WF-3) Consider the two normalized wave function shown below. Calculate the expectation value for…

A:

Q: Given a particle is confined to a 1D infinite potential well from -a/2 < x <a/2, prove that the…

A: Let us consider the Time independent Schrodinger equation in 1-dimension: -h22m∂2ψ∂x2+Vxψ=Eψ The…

Q: H.W.: prove that the normalization constant for the wavefunction (x) = A sin(=x) %3D 1-cos 20 Equal…

A: Given data, ψnx=Asinnπ2ax , where n is an integer.  which is a wave function of a particle in a…

Q: Consider a particle with 1-D wave-function (x) = kexp(-x²). Sketch (x),

A:

Q: A quantum mechanical particle is confined to a one-dimensional infinite potential well described by…

A: Step 1: Given: Particle in a 1-D infinite potential well described by the potential:V(x) =0,…

Q: A particle confined in a one-dimensional box of length L(&lt;= X &lt;= L) is in a state described by…

A:

Q: (b) Prove [A, BČ] =[Â, BJĈ + BLÂ, ĈJ- (c) Let the wave function for the particle is (r) ek Prove it…

A:

Q: Consider a potential in three regions: when x L, the potential is zero. Between 0 < x < L, the…

A:

Q: H.W.: prove that the normalization constant for the wavefunction , (x) = A sin(; 2a 1 1-cos 20,…

A: Solution

Q: A particle is confined to a 1D box between x=0 and x=1 and has the normalized wavefunction of 105 V…

A:

Q: b1 (x) = A sin () L

A:

Q: What would happen to the incident wave (x) = Ae¹kx at x < 0 that sees a step potential in the form…

A: a) Incident wave function is, ψx=Aeikx   at       x&lt;0 Potential is, Vx=iΓ Here, i is complex and…

Q: Consider a particle of mass m moving in one dimension with wavefunction $(x) Vi for sin L - and zero…

A: The momentum operator is given by Therefore the operator is given by Where The given wave…

Q: 3n s(2x – *), find 4normalized, the normalized wave function for a 1-dimensional particle- in-a-box…

A: Given wavefunction is, ψ=Acos2x-3π2 Here, A is the normalization constant. The normalization…

Q: (a) A non-relativistic, free particle of mass m is bouncing back and forth between two perfectly…

A: Quantum mechanics is a field of physics that studies the behavior of microscopic particles and their…

Q: In a scattering experiment, the potential is spherically symmetric and the particles are scat- tered…

A: The scattering amplitude is given by, f(θ)=1k∑l=0l=∞(2l+1)eiδlsinδlPlcosθ For the s wave, l=0 For…

Q: n partial wave analysis of scattering, one has to consider waves with L= 0, 1, 2, 3, For a given…

A: Concept used: Any particle getting scattered through potential is described as plane wave. In…

Q: ηπχ sin (1x). If L = 10.0, what is the L The eigenstates of the particle-in-a-box are written, n =…

A:

Q: Consider the wavefunction (4.5) with mi an integer and o < ¢< 2m. Find the normalization factor for…

A:

Q: Evaluate the transmission coefficient for a rectangular barrier with a potential given by Vo, (-a <x…

A:

Connection formulae for WKB approximation

In quantum mechanics, Schrödinger’s equation for certain systems with varying parameters cannot be solved directly. In such a case, an approximation method is used to obtain the approximate solution. WKB method is a special technique used to obtain the approximate solution for Schrödinger’s time-independent equation. In general, the WKB method is used to solve linear differential equations with varying coefficients. This method is called semiclassical approximation in quantum mechanics.