Physical meaning of the 2 eigenfunctions of Free Particle

[cks]

For Schrödinger's equation

$$\frac{\d^2\psi}{dx^2} = - \frac{2mE}{\hbar^2}\psi$$

Solving to find that

$$\psi = Aexp(ikx)+Bexp(-ikx)$$

I am curious about the physical meanings of the two terms of the solutions.

In solving a free particle encountering a potential barrier, In the region before the encounter of the barrier, the solutions of the Shcrodinger equation is just the free particle equation above. My teacher says the term with the positive sign means it's a wave going towards the barrier, whereas the negative signs is the wave that reflect from it.

Well, the wave function is just the solution of the Schrödinger's equation and how does my teacher derives the physical meaning from it?? I mean the exponential function has complex term, which are actually sinusoidal but doesn't tell us anything about the direction of going??

 

[vanesch]

For Schrödinger's equation

$$\frac{\d^2\psi}{dx^2} = - \frac{2mE}{\hbar^2}\psi$$

Solving to find that

$$\psi = Aexp(ikx)+Bexp(-ikx)$$

I am curious about the physical meanings of the two terms of the solutions.

In solving a free particle encountering a potential barrier, In the region before the encounter of the barrier, the solutions of the Shcrodinger equation is just the free particle equation above. My teacher says the term with the positive sign means it's a wave going towards the barrier, whereas the negative signs is the wave that reflect from it.

Well, the wave function is just the solution of the Schrödinger's equation and how does my teacher derives the physical meaning from it?? I mean the exponential function has complex term, which are actually sinusoidal but doesn't tell us anything about the direction of going??

That is because the operator of MOMENTUM, being P = hbar/i d/dx, gives you, applied to the first term:

P {exp(ikx) } = hbar k exp(ikx), meaning that the first term is an eigenfunction of the momentum operator with eigenvalue hbar k.

P {exp(- ikx) } = - hbar k exp(-ikx), meaning that the second term is an eigenfunction of the momentum operator with eigenvalue - hbar k.

So the first wavefunction represents also a state with momentum + hbar k,
while the second wavefunction represents a state with momentum - hbar k.

 

[Hootenanny]

Although I wouldn't worry about it for your course, you should perhaps at least be aware that your solution for a free particle is not the true solution, since it isn't square integrable. You'll see why if you do more advance QM courses.

 

[cks]

I see. The reason that a momentum operator acts on the eigenfunction produces a positive eigenvalue means its direction is towards the barrier.

thanks .

 

[vanesch]

I see. The reason that a momentum operator acts on the eigenfunction produces a positive eigenvalue means its direction is towards the barrier.

Well, it means that your momentum is a positive number. Depends then on how your axes are defined and where the barrier is of course...