# Superposition of states and orthonormalization

• tarletontexan
In summary, the problem posed is asking for the value of C necessary to normalize a particle in a superposition of states given by Y(x,0)=C(y1(x)+y2(x)), where y1(x) and y2(x) are stationary states with energies E1 and E2 respectively. The orthonormalization of y1(x) and y2(x) is causing confusion for the individual attempting to solve the problem. They inquire about the equation expressing the requirement for Y(x,0) to be normalized, and the definition of a normalized wavefunction. They also wonder if it is possible to get the given wavefunction into a similar format as an example in their textbook.
tarletontexan

## Homework Statement

Consider a particle in a superposition of states given at time t=0 by Y(x,0)=C(y1(x)+y2(x)), where y1(x) and y2(x) are the stationary states with energies E1 and E2 respectively. if y1(x) and y2(x) are orthonormalized, what value of C is required to normalize Y(x,0)?

## The Attempt at a Solution

the orthonormalization is messing with my head, there is an example of normalizing wavefunctions in my book but this looks nothing like it. I am not sure of how to approach this problem.

What equation expresses the requirement that Y(x,0) is normalized?

that is not given, the only thing it asks for in this part of the problem is what value of C is required to normalize Y(x,0)...

I realize it is not given in the problem statement. It should be given in your lecture notes or textbook. What does it mean, for a wavefunction to be normalized? Use the definition of "normalized", and apply it to this problem.

ok, i see, but the example in the book has the initial wavefunction given as Y(x,0)=Cexp(-|x|/x0) where C and x0 are constants, is it possible to get this function into a similar format??

## 1. What is the concept of superposition of states?

Superposition of states is a fundamental principle in quantum mechanics, which states that a quantum system can simultaneously exist in multiple possible states until it is observed or measured. This means that the system exists in a combination of all possible states at the same time, rather than being in a single specific state.

## 2. How does superposition of states relate to the wave-particle duality?

The concept of superposition of states is closely related to the wave-particle duality of quantum particles. This duality suggests that particles can exhibit both wave-like and particle-like behavior, and the superposition of states allows for this behavior by allowing a particle to exist in multiple states at once.

## 3. What is the significance of orthonormalization in quantum mechanics?

Orthonormalization is the process of making a set of states or vectors in a vector space orthogonal (perpendicular) to each other and normalized (having a magnitude of 1). In quantum mechanics, orthonormalization is important because it allows for the precise mathematical description of quantum states and their relationships with each other.

## 4. How is orthonormalization used in calculating probabilities in quantum mechanics?

In quantum mechanics, the probability of a quantum system being in a particular state is given by the square of the magnitude of the coefficient for that state in the state vector. Orthonormalization ensures that the state vector is normalized, so the sum of all probabilities for all possible states is equal to 1.

## 5. Can the superposition of states and orthonormalization be observed in everyday life?

No, the principles of superposition and orthonormalization are only applicable to the microscopic world of quantum mechanics. In everyday life, objects behave according to classical mechanics, where the concept of superposition does not apply. However, some technologies, such as quantum computers, utilize these principles to perform complex calculations and simulations.

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