Solution to time dependent wave equation

In summary, the given equation Y(x,t) = cos(kx)exp(-iwt) is a solution to the time-dependent Schrodinger wave equation, with k as the wavenumber and w as the angular frequency. The Hamiltonian of Y(x,t) = ihbar d/dt Y(x,t) is used to find the value for (hbar / 2m)*k^2, which is then equated to w in order to solve for the relation between wavenumber and angular frequency. The confusion is resolved by considering the equations for wavenumber and angular frequency, and the assumption that the energy component for momentum is entirely kinetic for a free particle.
  • #1
jaejoon89
195
0

Homework Statement



Show that Y(x,t) = cos(kx)exp(-iwt) is a solution to the time-dependent Schrodinger wave equation.

where k is the wavenumber and w is the angular frequency

Homework Equations



Hamiltonian of Y(x,t) = ihbar d/dt Y(x,t)

The Attempt at a Solution



When I plug everything in and do the calculus, I get

(hbar / 2m)*k^2 = w

I've been paying around with it but for the life of me cannot figure out how the left hand side is equal to w even though the unit check suggests it could. So my question is, how can you simplify the left hand side to w

(hbar / 2m)*k^2 = ?
 
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  • #2
Isn't there some relation between wavenumber and angular frequency given in your textbook?:wink:
 
  • #3
I can't find it. I know angular frequency is 2pi*nu and wavenumber is one over lambda. And I know nu*lambda = c, if that's the equation you mean, but still cannot get it to work...!

(hbar / 2m)(1/lambda^2) = (h/(4pi*m*lambda^2)) =...?
 
  • #5
Thanks! I got it now. A quick follow up: why is it assumed that the energy component for momentum (related to the wavenumber in this problem) is entirely kinetic?

i.e.,

p = mv => p^2 = m^2 v^2 = 2m (1/2) m v^2 = 2m E => p = sqrt(2mE)
 
  • #6
A free particle (I assume your problem is for a free particle, although you didn't actually say so in your problem statement) is subject to zero external forces/potentials, so its energy (in non-relativistic QM) is entirely kinetic.
 

1. What is the time dependent wave equation?

The time dependent wave equation is a mathematical formula used to describe the behavior of a wave over time. It is a partial differential equation that relates the second derivative of a wave function with respect to time to its spatial derivatives.

2. What does the solution to the time dependent wave equation represent?

The solution to the time dependent wave equation represents the amplitude and phase of a wave at a given point in space and time. It can be used to predict the future behavior of a wave based on its current state.

3. How is the time dependent wave equation used in different fields of science?

The time dependent wave equation has a wide range of applications in various fields of science, such as physics, engineering, and geology. It is used to study phenomena such as sound waves, electromagnetic waves, and seismic waves.

4. What are the key assumptions made in the time dependent wave equation?

The time dependent wave equation assumes that the medium in which the wave propagates is homogeneous, isotropic, and continuous. It also assumes that there are no external forces acting on the wave.

5. What are the limitations of the solution to the time dependent wave equation?

The solution to the time dependent wave equation is limited to linear systems, where the wave function is directly proportional to the external forces acting on it. It also does not account for phenomena such as dispersion, absorption, and nonlinearity.

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