Particle in one dimension wave function from Quantum Mechanics

In summary, the wave function of a particle in one dimension in quantum mechanics describes the probability amplitude of finding the particle at a given position and time. It is represented as a complex-valued function, typically denoted by Ψ(x, t), which evolves according to the Schrödinger equation. The square of the wave function's absolute value, |Ψ(x, t)|², gives the probability density of locating the particle at position x at time t. Boundary conditions and potential energy influence the form of the wave function, leading to discrete energy levels in bound systems, such as in quantum wells or harmonic oscillators. The wave function encapsulates the fundamental principles of superposition and uncertainty, reflecting the probabilistic nature of quantum mechanics.
  • #1
BlondEgg
4
1
Homework Statement
Using various def's in Quantum Mechanics for a particle in one dimension.
Relevant Equations
Wave function
Hi,

I try and solve this problem

1721511249924.png

I have solved the problem in different parts

1721511357658.png

1721511427351.png

But me not sure how to plot the graph. Maybe someone knows?

Merci
 
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  • #2
BlondEgg said:
But me not sure how to plot the graph. Maybe someone knows?
With a plotting program? Excel has the ERF function that you can use.
 
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  • #3
kuruman said:
With a plotting program? Excel has the ERF function that you can use.
merci

I plotted in overleaf latex

1721513497875.png

Do you know if the other parts (i) (ii) (iii) (iv) are good?

Best wishes to you
 
  • #4
BlondEgg said:
Do you know if the other parts (i) (ii) (iii) (iv) are good?
What do you think? There comes a time when you need to troubleshoot your own work. Look at your answers and try to prove them wrong. If you can't, then they are probably correct.

For example, in (i) when you integrated did you get 1 or did you not?
In (ii), is you answer the magnitude-squared of what you got in (i) or is it not?

And so on.
 
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FAQ: Particle in one dimension wave function from Quantum Mechanics

What is the wave function of a particle in one dimension?

The wave function of a particle in one dimension is a mathematical description that encapsulates the quantum state of the particle. It is typically represented as Ψ(x, t), where x is the position and t is time. The square of the absolute value of the wave function, |Ψ(x, t)|², gives the probability density of finding the particle at position x at time t.

How is the wave function related to the probability of finding a particle?

The wave function is directly related to the probability of locating a particle in a given region of space. Specifically, the probability of finding the particle between positions a and b is given by the integral of the square of the wave function over that interval: P(a ≤ x ≤ b) = ∫ab |Ψ(x, t)|² dx. This means that the wave function must be normalized so that the total probability over all space equals one.

What are the boundary conditions for the wave function?

Boundary conditions for the wave function are essential to ensure that the solutions to the Schrödinger equation are physically meaningful. Common conditions include requiring that the wave function be continuous and differentiable, and that it approaches zero as the position approaches infinity. For confined systems, such as a particle in a box, the wave function must also satisfy specific values at the boundaries (e.g., being zero at the walls of the box).

What is the significance of the time-dependent Schrödinger equation?

The time-dependent Schrödinger equation describes how the wave function of a quantum system evolves over time. It is a fundamental equation in quantum mechanics that allows us to predict the future state of a particle based on its initial wave function. The equation is given by iħ∂Ψ/∂t = - (ħ²/2m) ∂²Ψ/∂x² + V(x)Ψ, where ħ is the reduced Planck's constant, m is the mass of the particle, and V(x) is the potential energy as a function of position.

What are stationary states and how do they relate to the wave function?

Stationary states are specific solutions to the time-independent Schrödinger equation that describe quantum states with a definite energy. In these states, the probability density associated with the wave function does not change over time, although the wave function itself may acquire a time-dependent phase factor. Stationary states are important because they represent the allowed energy levels of a quantum system, and they can be used to construct the general solution of the wave function through superposition.

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