Wave function of a simple harmonic oscillator

In summary, the probability of the energy being measured and the system being in the ground state can be calculated by taking the integral of the wave function from 0 to infinity and dividing it by the integral of the ground state wave function from 0 to infinity. This will result in a function dependent on energy, which can be used to determine the probability at any given time.
  • #1
noblegas
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Homework Statement


The ground state wave function of a one-dimensional simple harmonic oscillator is

[tex]\varphi_0(x) \propto e^(-x^2/x_0^2)[/tex], where [tex]x_0[/tex] is a constant. Given that the wave function of this system at a fixed instant of time is [tex] \phi\phi \propto e^(-x^2/y^2)[/tex] where y is another constant., find the probablity, that if the energy is measured , the system will be in the ground state


Homework Equations





The Attempt at a Solution



[tex] dP=|\varphi_0|^2 dx[/tex]

According to my book(Peebles) , [tex]=|\varphi_0|^2=|\phi_0|^2[/tex];, so therefore [tex] dP=|\phi_0|^2 dx[/tex]?
 
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  • #2
anyone still not understand my question?
 
  • #3
hello noble,
i'm wondering if you take the integral of your state function from 0 to infinity then divide that result the integral of your ground state function from 0-infinity will you get a function dependent on energy, whereas you could use an energy sample at any time to give the probability?
 

1. What is a simple harmonic oscillator?

A simple harmonic oscillator is a type of system that exhibits periodic motion, meaning it oscillates back and forth between two points. It is characterized by its restoring force, which is directly proportional to its displacement from equilibrium.

2. What is the wave function of a simple harmonic oscillator?

The wave function of a simple harmonic oscillator describes the probability of finding the oscillator in a particular state. It is a mathematical function that varies over time and space, and it can be used to determine the energy levels and other properties of the oscillator.

3. How is the wave function of a simple harmonic oscillator derived?

The wave function of a simple harmonic oscillator can be derived using the Schrödinger equation, which describes how quantum systems evolve over time. By solving this equation for the simple harmonic oscillator, we can obtain the mathematical expression for its wave function.

4. What is the significance of the wave function in quantum mechanics?

The wave function is a fundamental concept in quantum mechanics, as it describes the behavior and properties of quantum systems. It contains all the information about a system that can be known, and it allows us to make predictions about the behavior of particles at the atomic and subatomic level.

5. How does the wave function change over time for a simple harmonic oscillator?

The wave function of a simple harmonic oscillator changes over time in a predictable way, as the oscillator oscillates between two points. The amplitude of the wave function increases and decreases, while its phase remains constant. This behavior is described by the Schrödinger equation and is a fundamental aspect of quantum mechanics.

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