Question 4What is the time-dependent Schrödinger equation for a particle in a potential 𝑉=12𝑚𝜔2𝑥2V= 21​ mω 2 x 2 ?1 point𝑖ℏ∂𝜓∂𝑡=12𝑚𝜔2𝑥2iℏ ∂t∂ψ​ = 21​ mω 2 x 2 𝑖ℏ∂𝜓∂𝑡=−ℏ2∇22𝑚𝜓+12𝑚𝜔2𝑥2𝜓iℏ ∂t∂ψ​ =− 2mℏ 2 ∇ 2 ​ ψ+ 21​ mω 2 x 2 ψ(𝑖ℏ∂∂𝑡+12𝑚𝜔2𝑥2)𝜓=𝐸𝜓(iℏ ∂t∂​ + 21​ mω 2 x 2 )ψ=Eψ𝑖ℏ∂𝜓∂𝑡=−ℏ2∇22𝑚𝜓+12𝑚𝜔2𝑥2iℏ ∂t∂ψ​ =− 2mℏ 2 ∇ 2 ​ ψ+ 21​ mω 2 x 2

Which of the following wavefunctions satisfy the time-dependent Schrödinger equation for a free particle? (Check all that apply)1 point𝜓(𝑥⃗,𝑡)=cos⁡(1ℏ(𝑝⃗⋅𝑥⃗−𝑝22𝑚𝑡))ψ( x ,t)=cos( ℏ1​ ( p​ ⋅ x − 2mp 2 ​ t))𝜓(𝑥⃗,𝑡)=sin⁡(1ℏ(𝑝⃗⋅𝑥⃗−𝑝22𝑚𝑡))ψ( x ,t)=sin( ℏ1​ ( p​ ⋅ x − 2mp 2 ​ t))𝜓(𝑥⃗,𝑡)=sin⁡(1ℏ(𝑝⃗⋅𝑥⃗+𝑝22𝑚𝑡))ψ( x ,t)=sin( ℏ1​ ( p​ ⋅ x + 2mp 2 ​ t))𝜓(𝑥⃗,𝑡)=exp⁡(𝑖ℏ(𝑝⃗⋅𝑥⃗−𝑝22𝑚𝑡))ψ( x ,t)=exp( ℏi​ ( p​ ⋅ x − 2mp 2 ​ t))exp⁡(𝑖ℏ𝐸𝑡)exp( ℏi​ Et) for (𝐸>0)(E>0)

Schrodinger wave equation

State the Schrodinger equation for a particle moving in one dimensionand show how it is related to the motion of electrons in atoms

A particle moves according to the equation; x = 10t2,where x is in meters and t is in seconds. Find the velocity forthe time interval from 2.0 s to 2.1 s

(Schrodinger's Equations - STIE and STDE