Real Number Expectation Value: Explanation & QM Example

  • I
  • Thread starter dyn
  • Start date
In summary, The conversation discusses a QM example and the expectation value for momentum. The equations include complex numbers α and β, and real numbers ω and t. It is mentioned that the two parts in the bracket are complex conjugates, leading to a real number when multiplied by -i. The speaker expresses gratitude for the explanation.
  • #1
dyn
773
61
hi. I have been looking at a QM example in a book. It works out the expectation value for momentum which I know should be a real number but I can't see how or why the following number is real
α and β are complex numbers with | α |2 + | β |2 = 1 and ω and t are real

-i ( α*β e-iωt -αβ*eiωt)

Can anyone tell me if this is a real number and why?
Thanks
 
Mathematics news on Phys.org
  • #2
The two parts in the bracket are the complex conjugates of each other: They have the same real component but opposite imaginary component. Their difference has a real component of zero, it is purely imaginary. If you multiply that by -i you get a real number.
 
  • Like
Likes QuantumQuest, Delta2 and dyn
  • #3
Thanks. I just couldn't see it but now I can. Thanks again
 

1. What is a real number expectation value?

A real number expectation value is a concept in quantum mechanics that represents the average value of a physical quantity, such as position or energy, that can be measured in a specific quantum state. It is calculated by taking the sum of all possible outcomes of a measurement, weighted by their probabilities.

2. How is the real number expectation value calculated?

The real number expectation value is calculated by taking the integral of the product of the wave function and the operator corresponding to the physical quantity being measured. This integral is then squared, and the result is multiplied by the complex conjugate of the wave function and integrated over the entire range of the variable being measured.

3. What is the significance of the real number expectation value in quantum mechanics?

The real number expectation value is significant because it represents the most probable result of a measurement in a given quantum state. It also helps to determine the uncertainty in the measurement, as the difference between the expectation value and the actual measurement can be used to calculate the standard deviation of the measurement.

4. Can the real number expectation value be negative?

Yes, the real number expectation value can be negative. This indicates that there is a non-zero probability of obtaining a negative value for the measured physical quantity. However, in some cases, the expectation value may be negative due to the mathematical properties of the wave function and operator used in the calculation.

5. What is an example of calculating the real number expectation value?

An example of calculating the real number expectation value is determining the average energy of a particle confined to a one-dimensional box. The expectation value can be found by taking the integral of the product of the wave function and the energy operator, which is the second derivative of the wave function. The resulting value represents the average energy of the particle in the given quantum state.

Similar threads

I Newton-Raphson Method With Complex Numbers
    • General Math
Replies
7
Views
1K
B Getting from complex domain to real domain
    • General Math
Replies
3
Views
809
Better definition for complex number
    • General Math
2
Replies
41
Views
4K
I Applications of complex gamma (or beta) functions in physics?
    • General Math
Replies
6
Views
2K
MHB Property of real-valued Fourier transformation
    • General Math
Replies
1
Views
1K
A Explicit example of two complex numbers for double cover U(1) of SO(2) for a specified angle
    • General Math
Replies
2
Views
401
Quantum - Two State Problem in different bases
    • Advanced Physics Homework Help
Replies
3
Views
1K
B Multiterm Taylor expansion
    • Calculus
Replies
1
Views
1K
I Domain of single-valued logarithm of complex number z
    • General Math
Replies
2
Views
2K
Summing sinusoids of different frequency
    • General Math
Replies
9
Views
25K

Hot Threads

Recent Insights

Back
Top