Normalised Energy Eigenfunction (Probability with Dirac Notation)

In summary, the conversation discusses the normalised energy eigenfunction for the ground state of a one-dimensional harmonic oscillator and a prepared state for the oscillator. The question is about finding the probability of a measurement of the energy giving a specific result. The solution involves squaring the states to get probabilities and using the definition of probability to find the scalar product of the two states involved. The integration of the resulting complex number is then done, using the typical gaussian integration, to find the probability.
  • #1
Unto
128
0

Homework Statement



Normalised energy eigenfunction for ground state of a harmonic oscillator in one dimension is:

〈x|n〉=α^(1/2)/π^(1/4) exp(-□(1/2) α^2 x^2)

n = 0

α^2=mω/h

suppose now that the oscillator is prepared in the state:

〈x|ψ〉=σ^(1/2)/π^(1/4) exp(-(1/2) σ^2 x^2)


What is the probability that a measurement of the energy gives the result E_0 = 1/2 hω?


Homework Equations





The Attempt at a Solution



I squared both states as that is the probability of the states, but I cannot see where the energy eigenvalue would fit into it.

It probably something glaringly obvious, but I haven't done this in a long time and certainly not with dirac notation. Please help.
 
Physics news on Phys.org
  • #2
I squared the states to get probabilities, but now I'm left with 2 probability amplitudes that I'm stumped on how to use.

What can I use the energy for?

I know n = 0, but do I have to find the expectation value of getting the energy?
 
  • #3
Apply the definition. The probability should be equal to the square modulul of some complex number, namely the scalar product of the 2 states involved. Which are the 2 states involved ?
 
  • #4
Since both states are real, their conjugates equal their normal counterparts. So I end up using an Identity relation to superimpose the states anyway (underhanded trick from my QM lecturer :p ).

Ok so I'm multiplying and integrating over unity, hopefully a useful co-efficient should pop out which I should then modulus square for the probability? Will show my working a hot second.
 
  • #5
How do I integrate e^[(-1/2)x^2] dx from -∞ to ∞ ?
 
  • #6
It's the typical gaussian integration. You should find its value in your QM book, or statistics book. Look it up.
 

1. What is a normalised energy eigenfunction?

A normalised energy eigenfunction is a mathematical function that describes the probability of finding a particle in a specific energy state. It is a solution to the Schrödinger equation and is represented using Dirac notation.

2. How is normalisation of an energy eigenfunction achieved?

Normalisation of an energy eigenfunction is achieved by ensuring that the integral of the squared magnitude of the function over all space is equal to 1. This represents the probability of finding the particle in any position and is necessary for the function to accurately describe the probability distribution.

3. What is the significance of the normalisation constant in an energy eigenfunction?

The normalisation constant in an energy eigenfunction is a crucial part of the function as it ensures that the total probability of finding the particle in any position is equal to 1. It is also used to calculate the probability of finding the particle in a specific energy state by taking the squared magnitude of the function at that energy level.

4. How does the probability of finding a particle change with different energy eigenfunctions?

The probability of finding a particle changes with different energy eigenfunctions because each function represents a different energy state. The higher the energy state, the larger the amplitude of the function, meaning there is a higher probability of finding the particle in that state. However, the overall probability of finding the particle in any energy state is always equal to 1 due to the normalisation constant.

5. Can the normalised energy eigenfunction be used to determine the exact position of a particle?

No, the normalised energy eigenfunction cannot be used to determine the exact position of a particle. It only describes the probability of finding the particle in a specific energy state and does not provide information about the particle's position. To determine the position of a particle, other mathematical functions, such as the wave function, must be used.

Similar threads

  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Quantum Physics
Replies
9
Views
2K
  • Advanced Physics Homework Help
Replies
1
Views
873
  • Advanced Physics Homework Help
Replies
1
Views
2K
  • Advanced Physics Homework Help
Replies
4
Views
1K
  • Advanced Physics Homework Help
Replies
14
Views
3K
  • Advanced Physics Homework Help
Replies
1
Views
1K
  • Advanced Physics Homework Help
Replies
3
Views
1K
  • Advanced Physics Homework Help
Replies
6
Views
5K
  • Advanced Physics Homework Help
Replies
7
Views
1K
Back
Top