Physical meaning of the 2 eigenfunctions of Free Particle

In summary, when solving Schrodinger's equation for a free particle encountering a potential barrier, the solutions are given by the equation \psi = Aexp(ikx)+Bexp(-ikx). The term with a positive sign represents a wave moving towards the barrier, while the term with a negative sign represents a wave reflecting from the barrier. This is because the first term is an eigenfunction of the momentum operator with a positive eigenvalue, indicating a direction towards the barrier, while the second term is an eigenfunction with a negative eigenvalue, indicating a direction away from the barrier. The use of complex exponential functions does not determine the direction, but rather the action of the momentum operator on the eigenfunctions gives insight into the direction
  • #1
cks
165
0
For Schrodinger's equation

[tex] \frac{\d^2\psi}{dx^2} = - \frac{2mE}{\hbar^2}\psi [/tex]

Solving to find that

[tex] \psi = Aexp(ikx)+Bexp(-ikx) [/tex]

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 Schrodinger'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??
 
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  • #2
cks said:
For Schrodinger's equation

[tex] \frac{\d^2\psi}{dx^2} = - \frac{2mE}{\hbar^2}\psi [/tex]

Solving to find that

[tex] \psi = Aexp(ikx)+Bexp(-ikx) [/tex]

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 Schrodinger'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 teh 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.
 
  • #3
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.
 
  • #4
I see. The reason that a momentum operator acts on the eigenfunction produces a positive eigenvalue means its direction is towards the barrier.

thanks .
 
  • #5
cks said:
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...
 

Related to Physical meaning of the 2 eigenfunctions of Free Particle

What is the physical meaning of the two eigenfunctions of a free particle?

The two eigenfunctions of a free particle represent the two possible states of the particle - the state of being in motion and the state of being at rest. These eigenfunctions are solutions to the Schrödinger equation and describe the probability distribution of finding the particle in a particular position at a given time.

How do the two eigenfunctions of a free particle differ from each other?

The two eigenfunctions of a free particle differ in terms of their momentum. The first eigenfunction represents a particle with zero momentum, also known as the state of rest. The second eigenfunction represents a particle with non-zero momentum, also known as the state of motion. This is because the momentum operator is used to obtain these eigenfunctions, and the two eigenfunctions are orthogonal to each other.

What is the significance of the eigenvalues associated with the two eigenfunctions of a free particle?

The eigenvalues associated with the two eigenfunctions of a free particle represent the possible energy levels of the particle. The first eigenfunction has an eigenvalue of zero, representing the lowest energy state of the particle - the state of rest. The second eigenfunction has a non-zero eigenvalue, representing higher energy states of the particle - the states of motion.

Why is it important to understand the physical meaning of the two eigenfunctions of a free particle?

Understanding the physical meaning of the two eigenfunctions of a free particle is crucial in understanding the behavior of quantum systems. These eigenfunctions represent the fundamental states of a particle and provide insights into its energy levels and probability distribution. This understanding is essential in many areas of physics, including quantum mechanics and statistical mechanics.

How are the two eigenfunctions of a free particle related to the Uncertainty Principle?

The two eigenfunctions of a free particle are related to the Uncertainty Principle in that they represent the two complementary properties of a particle - position and momentum. The eigenfunction of the state of rest has a well-defined position but an uncertain momentum, while the eigenfunction of the state of motion has a well-defined momentum but an uncertain position. This is a manifestation of the Uncertainty Principle, which states that it is impossible to simultaneously know the exact values of these two properties of a particle.

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