# Introductory QM boundary conditions

• stephen8686
In summary, the wave function representing a particle has the condition that it must be continuous at the points -L/2, 0, and +L/2. To find the value of C, the normalization condition is used, which states that the probability of finding the particle somewhere between x = -∞ and x = +∞ must equal 1. This leads to the conclusion that C = 0, as at x = L/2 the value of C does not affect the equation.
stephen8686

## Homework Statement

A particle is represented by the following wave function:
ψ(x)=0 x<-L/2
=C(2x/L+1) -L/2<x<0
=C(-2x/L+1) 0<x<+L/2
=0 x>+L/2

use the normalization condition to find C

## Homework Equations

ψ(x) must be continuous[/B]

## The Attempt at a Solution

I'm supposed to say that at the points -L/2, 0, +L/2 ψ(x) must be continuous, so then I can find C. But I get C=0. For example at x= L/2 I get 0=0 and 0= -2C/L for the conditions so C=0.
I feel like I must just either overlooking something stupid or I'm just doing this completely wrong.

stephen8686 said:
1But I get C=0. For example at x= L/2 I get 0=0 and 0= -2C/L for the conditions so C=0.
I feel like I must just either overlooking something stupid or I'm just doing this completely wrong.
At x = L/2 you get 0 = 0 no matter what the value of C. So, this doesn't help you determine C.

Find C by using the idea that the probability must equal 1 for finding the particle somewhere between x = -∞ and x = +∞.

stephen8686
TSny said:
At x = L/2 you get 0 = 0 no matter what the value of C. So, this doesn't help you determine C.

Find C by using the idea that the probability must equal 1 for finding the particle somewhere between x = -∞ and x = +∞.

Thanks TSny, I think I got it now

## 1. What are boundary conditions in introductory quantum mechanics?

Boundary conditions in introductory quantum mechanics refer to the conditions that must be satisfied at the boundaries of a quantum system. These conditions dictate the behavior of the system at its boundaries and are crucial for understanding the properties and dynamics of the system.

## 2. What types of boundary conditions are commonly encountered in introductory quantum mechanics?

There are two types of boundary conditions commonly encountered in introductory quantum mechanics: boundary conditions for wave functions and boundary conditions for operators. Wave function boundary conditions specify the behavior of the wave function at the boundaries of a system, while operator boundary conditions dictate the behavior of operators (such as position and momentum) at the boundaries.

## 3. Why are boundary conditions important in quantum mechanics?

Boundary conditions are important in quantum mechanics because they help determine the allowed energy states and the probability of finding a particle in a given location. They also play a crucial role in solving the Schrödinger equation, which describes the time-evolution of quantum systems.

## 4. How are boundary conditions applied in introductory quantum mechanics?

In introductory quantum mechanics, boundary conditions are typically applied by solving the Schrödinger equation with the given boundary conditions. This results in a set of allowed energy states and corresponding wave functions for the system. These solutions can then be used to calculate various properties of the system, such as the probability of finding a particle in a given location.

## 5. Can boundary conditions change the behavior of a quantum system?

Yes, boundary conditions can significantly impact the behavior of a quantum system. For example, different boundary conditions can lead to different energy states and wave functions, resulting in different probabilities for finding a particle in a given location. Additionally, changing the boundary conditions can alter the quantization of a system, leading to different physical properties and behaviors.

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