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)

The time-dependent Schrödinger equation for a free particle is given by:

iℏ ∂ψ/∂t = -ℏ²/2m ∇²ψ

The solutions to this equation are wavefunctions of the form:

ψ(x,t) = A exp[i(kx - ωt)]

where A is the amplitude of the wave, k is the wave number, x is the position, ω is the angular frequency, and t is time. The wave number and angular frequency are related to the momentum and energy of the particle by the relations p = ℏk and E = ℏω.

Let's check each of the given wavefunctions:

1. ψ(x,t) = cos(1/ℏ(p⋅x - p²/2mt)): This is not a solution because it is not of the form A exp[i(kx - ωt)].

2. ψ(x,t) = sin(1/ℏ(p⋅x - p²/2mt)): This is not a solution for the same reason as above.

3. ψ(x,t) = sin(1/ℏ(p⋅x + p²/2mt)): This is not a solution for the same reason as above.

4. ψ(x,t) = exp(i/ℏ(p⋅x - p²/2mt)): This is a solution because it is of the form A exp[i(kx - ωt)].

5. exp(iℏEt) for (E>0): This is not a solution because it does not depend on the position x.

So, the only wavefunction that satisfies the time-dependent Schrödinger equation for a free particle is ψ(x,t) = exp(i/ℏ(p⋅x - p²/2mt)).