The postulate of Quantum Mechanics and Eigenvalue equation

In summary: The probability that a measurement of position will yield the specific eigenvalue q is ##\frac{1}{2}##.In summary, according to the postulate, every measured observable is an eigenvalue of a corresponding linear Hermitian operator. This means that if you want to know the exact energy levels of a system, you need to measure its position. However, because the system state is not necessarily an eigenstate before you measure, there is always some uncertainty about the resulting value.
  • #1
betelgeuse91
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According to one of the postulates of quantum mechanics, every measured observable q is an eigenvalue of a corresponding linear Hermitian operator Q. Which means, that q must satisfy the equation Qψ = qψ. But according to Griffiths chapter 3, this equation can only be followed from σQ = 0. It makes no sense to me because not every observable has zero standard deviation. Can someone explain this?
 
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The exact energy levels (ferinstance) are for idealized situations... real energy levels are best characterised by a line with some thickness appropriate to the uncertainty of the state - which leads to it's stability. So you are learning that the postulates are not exactly true.

You also need to distinguish between uncertainties in the Heisenberg sense and measurement uncertainties arising from the way measurement equipment works.
Real life is messy.
 
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  • #3
I am only considering idealized theoretical situations. The position of harmonic oscillator, for example, according to the postulate, a measured position value q is an eigenvalue of the equation xψ = qψ. But from this equation we can deduce that σx = 0. This is not true since |ψ|2 is not a single valued function. I am so confused...
 
  • #4
betelgeuse91 said:
I am only considering idealized theoretical situations. The position of harmonic oscillator, for example, according to the postulate, a measured position value q is an eigenvalue of the equation xψ = qψ. But from this equation we can deduce that σx = 0. This is not true since |ψ|2 is not a single valued function. I am so confused...

The postulate says that if you measure a given observable, the result will be one of the eigenvalues of that observable. However, the system state is not necessarily an eigenstate before you measure; it may be a superposition of several different eigenstates with different eigenvalues and then your measurement may yield any of several different results. Formally, you prepare an ensemble of systems all in the same initial state and measure the observable on each instance. If the initial state is an eigenstate of the observable, you will get the corresponding eigenvalue on every measurement and ##\sigma## will be zero. However, if the initial state is a superposition you will get different results (all eigenvalues of one of the many eigenstates making up the superposition) on the different measurements and ##\sigma## will be non-zero.

For a number of reasons, it is not possible to prepare a system in an exact eigenstate of the position operator, so ##\sigma_x## will always be non-zero.
 
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  • #5
betelgeuse91 said:
Can someone explain this?
The standard deviation depends on the state. In an eigenstate of ##Q##, there is no uncertainty about the value, so the standard deviation is zero, as it should be.
 
  • #6
betelgeuse91 said:
I am only considering idealized theoretical situations. The position of harmonic oscillator, for example, according to the postulate, a measured position value q is an eigenvalue of the equation xψ = qψ. But from this equation we can deduce that σx = 0. This is not true since |ψ|2 is not a single valued function. I am so confused...
What they said... plus: what is the probability that a measurement of position will yeild the specific eigenvalue q? Explain how you worked it out so I can see your current understanding and I'll get back to you.

Note: the position operator has continuous eigenvalues, as does momentum.
The particular value for position is one of the allowed eigenvalues.
 

1. What is the postulate of quantum mechanics?

The postulate of quantum mechanics states that the state of a quantum system can be described by a wave function, which evolves according to the Schrödinger equation. It also states that the measurement of a physical quantity corresponds to an eigenvalue of the corresponding operator acting on the wave function.

2. What is an eigenvalue equation in quantum mechanics?

An eigenvalue equation in quantum mechanics is an equation that relates the eigenvalues and eigenvectors of an operator to the wave function of a quantum system. It is used to calculate the probabilities of different outcomes of a measurement on a quantum system.

3. How is the eigenvalue equation related to the uncertainty principle?

The eigenvalue equation is related to the uncertainty principle in that the eigenvalues of a physical quantity, such as position or momentum, correspond to the possible outcomes of a measurement. The uncertainty principle states that it is impossible to know both the exact position and momentum of a particle at the same time, and the eigenvalue equation helps to determine the probabilities of these outcomes.

4. How does the postulate of quantum mechanics differ from classical mechanics?

The postulate of quantum mechanics differs from classical mechanics in that it describes the behavior of particles at the quantum level, where classical mechanics breaks down. It takes into account concepts such as wave-particle duality and the uncertainty principle, which are not present in classical mechanics. Additionally, classical mechanics is deterministic, while quantum mechanics is probabilistic.

5. What are some applications of the postulate of quantum mechanics?

The postulate of quantum mechanics has many applications in various fields, such as quantum computing, quantum cryptography, and quantum chemistry. It also plays a crucial role in understanding the behavior of particles at the microscopic level, which has led to advancements in technology such as transistors and lasers. Additionally, it has allowed for the development of new materials and technologies, such as superconductors and MRI machines.

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