What Are the Conditions for a Particle in a Bound State of a Potential Well?

In summary, for a particle to be in a bound state of a potential well, the wave-function must be continuous, normalized, and tend to the classical limit. It must also be real-valued except for a trivial harmonic time dependence.
  • #1
bon
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Homework Statement



State three conditions that must be satisfied by the wave-function of a particle that is in a bound state of a potential well.


Homework Equations





The Attempt at a Solution



Not sure what the three are!?
I can only think of one: the wavefunction must be continuous.

The derivative doesn't necessarily need to be continuous does it? e.g. in delta function potential well it isnt..

Also it doesn't need to vanish at the ends of the well does it--this is only for an infinite well..

so what are the other two?


Thanks!
 
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  • #2
bon said:

Homework Statement



State three conditions that must be satisfied by the wave-function of a particle that is in a bound state of a potential well.


Homework Equations





The Attempt at a Solution



Not sure what the three are!?
I can only think of one: the wavefunction must be continuous.

The derivative doesn't necessarily need to be continuous does it? e.g. in delta function potential well it isnt..

Also it doesn't need to vanish at the ends of the well does it--this is only for an infinite well..

so what are the other two?


Thanks!
The only other ones I can think of is the wave-function must be normalized so the total probability is 1, and that it has to tend to the classical limit.
 
  • #3
Bound states are stationary states, meaning that apart from a trivial harmonic time dependence, the wavefunction does not change shape over time. What this also means is that apart from the harmonic time factor, the wavefunction is real-valued, whereas wavefunctions in general are complex-valued.
 

1. What is a wavefunction and how is it related to quantum mechanics?

A wavefunction is a mathematical function that describes the quantum state of a system. It contains information about the position, momentum, and energy of particles in the system. It is a fundamental concept in quantum mechanics and helps to explain the behavior of particles at the atomic and subatomic level.

2. What are the main properties of a wavefunction?

The main properties of a wavefunction include being continuous, single-valued, and square-integrable. It must also be normalized, meaning that the total probability of finding a particle in a given space is equal to 1. Additionally, a wavefunction must satisfy the Schrödinger equation, which describes how the wavefunction evolves over time.

3. How is the probability of finding a particle related to the wavefunction?

The probability of finding a particle in a specific location is related to the square of the wavefunction. This is known as the Born rule, which states that the probability density of finding a particle in a certain location is proportional to the square of the magnitude of the wavefunction at that point.

4. Can the wavefunction of a particle be observed directly?

No, the wavefunction itself cannot be observed directly. Instead, scientists use mathematical models and equations to describe the behavior of particles at the quantum level. The wavefunction is a mathematical concept that helps to predict the behavior of particles, but it cannot be directly measured.

5. How does the wavefunction collapse when a measurement is made?

When a measurement is made on a particle, its wavefunction collapses to a single state. This means that the particle's position, momentum, or other properties become definite and no longer exist in a superposition of multiple states. The exact mechanism of wavefunction collapse is still a topic of debate and research in quantum mechanics.

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