Understanding the Probability of Measuring an Allowed Energy in a Wave Function

In summary: No, measuring the wavelength of a photon emitted from an electron does not put the electron in a stationary state because the energy of the photon is always changing.
  • #1
DocZaius
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In Griffiths' page 36/37 he says "As we'll see in Chapter 3, what |Cn|^2 tells you is the probability that a measurement of the energy would yield the value En (a competent measurement will always return one of the "allowed" values - hence the name - and |Cn|^2 is the probability of getting the particular value En."

This statement comes at the end of the infinite square well section and it concerns any sum of stationary states.

Wouldn't measuring one of the allowed energies "En", mean collapsing the wave function into the nth stationary state (and that state only!)? Doesn't that mean that the wave function has then assumed a single stationary state, thereby guaranteeing all subsequent energy measurements of the wave function to yield that particular En?

This seems to disagree with my intuition about what happens to a wave function after measurement (that it would "delocalize" in energy space) and it disagrees with my professor's interpretation who literally takes issue with the statement that an energy measurement must yield one of the allowed values. He says that it can yield any value, and that a mix/sum of specific n states are responsible for returning a non-"allowed" En.
 
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  • #2
DocZaius said:
Wouldn't measuring one of the allowed energies "En", mean collapsing the wave function into the nth stationary state (and that state only!)? Doesn't that mean that the wave function has then assumed a single stationary state, thereby guaranteeing all subsequent energy measurements of the wave function to yield that particular En?

Yes.

DocZaius said:
This seems to disagree with my intuition about what happens to a wave function after measurement (that it would "delocalize" in energy space)

I always liked the saying, "the great thing about physical intuition is that it can be adjusted to fit the facts."

DocZaius said:
and it disagrees with my professor's interpretation who literally takes issue with the statement that an energy measurement must yield one of the allowed values. He says that it can yield any value, and that a mix/sum of specific n states are responsible for returning a non-"allowed" En.

The *expectation value* of energy can be a non-allowed value. This is the same as the fact that the expected (mean) value of a six-sided die is 3.5. But you'll always get an integer if you roll the die.
 
  • #3
So just to be clear - once I measure the energy of my infinite square well wave function, I have set it into a stationary state corresponding to that allowed energy and it will remain there subsequently? All future measurements, for all time, of the energy of that wave function will be my first energy measurement? Doesn't that violate some version of the uncertainty principle? The delta E delta t one?

This very much disagrees with my professor so I want to make sure! And I understand your point about the expectation value but we were talking about an actual single measurement.
 
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  • #4
DocZaius said:
So just to be clear - once I measure the energy of my infinite square well wave function, I have set it into a stationary state corresponding to that allowed energy and it will remain there subsequently? All future measurements, for all time, of the energy of that wave function will be my first energy measurement?

Yes, as long as you don't disturb the system by doing something other than an energy measurement.

DocZaius said:
Doesn't that violate some version of the uncertainty principle? The delta E delta t one?

No, it's a consequence of the energy-time uncertainty principle. Griffiths has an explanation of the meaning of this uncertainty relation; have you read it?

The idea is that (delta E)(delta T) ~ hbar, where (delta E) is the spread in energy and (delta T) is some measure of the characteristic time over which the system changes. If we reduce (delta E) to zero, then (delta T) must go to infinity; that is, it takes infinitely long for the system to undergo any changes; that is, the system never changes. Of course, this is why we call energy eigenstates "stationary states"!
 
  • #5
So all one needs to do to create a wave function that is exactly one stationary state is to create any wave function at all first (which is going to be a sum of stationary states), then measure its energy. Now you have a very simple single function of time and space rather than a sum of many. Can you give an example of an energy measurement in the real world which might be unperturbative enough to be a "pure" energy measurement?

What about measuring the wavelength of a photon emitted from an electron. If a specific allowed jump in energy of an electron caused this photon to be emitted, might not my measurement of that photon's wavelength be considered a "pure" energy measurement of the electron? Would that very measurement of the photon not put the electron in a stationary state? I am of course assuming that one could express the wave function of an electron as a sum of stationary states, which assumes that the time dependent Schrodinger equation of the electron (in some potential) would be separable, which is not necessarily true.
 
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  • #6
To prepare a photon in a highly precise energy state you can use an optical cavity: http://en.wikipedia.org/wiki/Optical_cavity

An optical cavity consists of a pair of partially transparent mirrors separated by a certain distance. If you shoot a photon at it, it will only pass through if it has a wavelength (thus energy) in a very narrow range. You can think of this as an energy measurement measurement than detects whether the photon has a certain energy or not.

A problem with thinking about stationary states of real-world electrons is that the only stationary state of an electron in an actual atom is the ground state. The excited states aren't pure energy eigenstates: they have a finite lifetime, after which they transition to the ground state + a photon, so they aren't stationary.
 
  • #7
You should make sure that you are not misunderstanding your prof. In any real world situation, the particle will not be perfectly isolated and there is no infinite well, so over time outside influences will cause the state to evolve from the stationary state. However, in the theoretical perfect world of the infinite square well, a perfect measurement will leave the particle in a fixed state.

Whenever real world situation come in, there will be many complications to this, but don't confuse those complications with the fundamental theory.

There are more complicated situations when it comes to a measurement of the energy, but a measurement that gives you zero uncertainty for the energy of a single particle will necessarily be an eigenvalue of the state.
 

1. What is a wave function?

A wave function is a mathematical representation of the quantum state of a particle. It describes the probability of finding a particle in a particular location or state.

2. What is an allowed energy in a wave function?

An allowed energy in a wave function refers to the possible energy levels that a particle can have in a given system. These energy levels are quantized, meaning they can only take on certain discrete values.

3. How is the probability of measuring an allowed energy calculated in a wave function?

The probability of measuring an allowed energy in a wave function is determined by the square of the amplitude of the wave function at that particular energy level. This is known as the Born rule.

4. What factors influence the probability of measuring an allowed energy in a wave function?

The probability of measuring an allowed energy in a wave function is influenced by several factors, including the shape of the wave function, the energy of the system, and any external forces acting on the particle.

5. Why is understanding the probability of measuring an allowed energy important in quantum mechanics?

In quantum mechanics, measurements of physical quantities, such as energy, are probabilistic rather than deterministic. This means that understanding the probability of measuring an allowed energy is crucial in predicting and interpreting the behavior of quantum systems.

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