Discrete and continuous confusion

In summary, quantum mechanics allows for discrete energy states of particles, but these states are not associated with specific positions. Rather, they are associated with certain wavefunctions, which can be either discrete or continuous.
  • #1
qbslug
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I don't understand how in quantum mechanics we have discrete and exact energy states for electron orbits but then at the same time we have a continuous probability density function for the position of an electron.
This seems like a paradox (although I know it can't be) since considering a continuous position distribution you would expect the different positions of the electron to create different energies of the electron in a continuous way. How is this explained? How can you have one energy for different positions?
 
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  • #2
Definite energies are not associated to each position state, but to certain wavefunctions called energy eigenstates (eigenvectors of the Hamiltonian operator). In certain situations, the energy spectrum associated with the energy eigenvectors is discrete. In other situations it may be continuous.
 
  • #3
May be is becouse you're thinking semi-classicaly like in Bhor's rule where only certain orbits are allowed while others are forbidden.
 
  • #4
qbslug said:
I don't understand how in quantum mechanics we have discrete and exact energy states for electron orbits but then at the same time we have a continuous probability density function for the position of an electron.

I don't think it is quite correct to say that we have "exact" discrete energy states for real electrons. A soon as you allow for coupling to the vacuum the energies are "smeared put" and the eigeneneriges are then just the centres of Lorentzian energy distributions.

It is pretty much analogues to resonances in an electrical LC circuit, in an ideal LC circuit there is a single discrete resonance frequency but as soon as you add some damping the resonance is broadened.

Note that this is just a consequences of the "mathematical uncertainty principle", in order to have an exact energy the electrons would have to stay in a single energy state for an infinite amount of time; a system with a finite lifetime can not have a truly discrete energy spectrum.
 

What is the difference between discrete and continuous data?

Discrete data is information that can only take on specific, distinct values. This type of data is often counted and represented by whole numbers. On the other hand, continuous data can take on any value within a range and is usually measured. This type of data is often represented by decimals or fractions.

How do you determine if data is discrete or continuous?

One way to determine if data is discrete or continuous is by looking at the scale of measurement. If the data can be measured and divided into smaller units, it is likely continuous. If the data can only be counted and cannot be divided into smaller units, it is likely discrete.

What are some examples of discrete data?

Examples of discrete data include the number of siblings a person has, the number of cookies in a jar, and the number of students in a classroom. These values can only take on specific, whole numbers and cannot be divided into smaller units.

What are some examples of continuous data?

Examples of continuous data include height, weight, temperature, and time. These values can take on any value within a range and can be measured in smaller units, such as inches, pounds, degrees, and seconds.

Why is it important to distinguish between discrete and continuous data?

It is important to distinguish between discrete and continuous data because they require different types of analysis and methods of interpretation. For example, discrete data is often analyzed using counts and percentages, while continuous data is often analyzed using measures of central tendency and variability. By correctly identifying the type of data, scientists can choose the appropriate statistical methods to analyze and draw conclusions from the data.

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