16 Questions
What does the wave function Ψ(x,t) in quantum mechanics represent?
The state of the system or information about observables
According to Born's interpretation, what does the probability density P(x,t) represent?
The probability per unit length of finding the particle near a specific coordinate at time t
Which physical parameters are considered as observables in quantum mechanics?
Energy, momentum, kinetic energy, and spin
What is the limitation on the accurate measurement of observables in quantum mechanics?
Limited by the uncertainty principle
What kind of information can be extracted from the wave function using appropriate operators?
Physical parameters like energy and spin
In quantum mechanics, what does a measurement on a system at the same state multiple times result in?
Consistent values or average values for observables if the state is not modified by the measurement
What is the general solution for the simple harmonic motion represented by the wave equation provided?
$\sin(kx) + \cos(kx)$
Based on the given conditions, what can be inferred about the value of B in the general solution?
$B = 0$
What does the quantization condition $kL = n\pi$ imply for the possible values of $k$ and $n$?
$k = \pi/nL$
What does the quantum number 'n' correspond to in terms of the energy levels?
First excited state
What are the eigenvalues of energy corresponding to a given energy level based on the provided equations?
$\frac{n^2\pi^2\hbar^2}{8m}$
What is the eigen function for a given quantum state 'n'?
𝛹𝑛 (𝑥) = 𝐴 𝑠𝑖𝑛(𝑛𝜋𝑥 ⁄𝐿)
How can the constant 'A' in the eigen function be evaluated?
From normalization condition
What does the probability density function 𝑃𝑛 (𝑥) represent?
Probability of finding the particle in a small region 'dx' in a quantum state 'n'
What trigonometric identity is used in the integration to evaluate the constant 'A'?
Sin2θ = (1 - cos2θ)/2
What does the eigen value 𝐸𝑛 represent?
Energy associated with the nth quantum state
Study Notes
Quantum Mechanics
- The wave equation is 𝑑²𝛹(𝑥)/𝑑𝑥² + 𝑘²𝛹(𝑥) = 0, representing simple harmonic motion.
- The general solution is 𝛹(𝑥) = 𝐴 sin 𝑘𝑥 + 𝐵 cos 𝑘𝑥.
- Applying the condition 𝛹(𝑥) = 0 at 𝑥 = 0, we get 𝐵 = 0.
- Substituting the condition 𝛹(𝑥) = 0 at 𝑥 = 𝐿, we get 𝐴 sin 𝑘𝐿 = 0, implying 𝑘𝐿 = 𝑛𝜋 or 𝑘 = 𝑛𝜋⁄𝐿.
- The quantization condition is 𝑘𝐿 = 𝑛𝜋, where 𝑛 = 1, 2, 3, ... is a positive integer.
Eigen Values of Energy
- The eigen values of energy are 𝐸𝑛 = ℎ²𝑘²/8𝜋²𝑚 = 𝑛²ℎ²/8𝜋²𝑚𝐿².
- 'n' is the quantum number corresponding to a given energy level, where n = 1 corresponds to the ground state, n = 2 corresponds to the first excited state, and so on.
Born's Interpretation
- In 1926, Max Born related the properties of the wave function 𝛹(𝑥, 𝑡) and the behavior of the associated particle, expressed in terms of the probability density 𝑃(𝑥, 𝑡).
- The probability density is 𝑃(𝑥, 𝑡) = 𝛹 ∗(𝑥, 𝑡)𝛹(𝑥, 𝑡), where 𝛹 ∗(𝑥) is the complex conjugate of 𝛹(𝑥, 𝑡).
Observables
- Observables are physical parameters associated with the particle, such as energy, momentum, kinetic energy, spin, etc.
- Experimental results can give us values of observables, and multiple measurements on the system at the same state should result in the same value or average values for the observables.
Eigen Functions
- The allowed wave functions are 𝛹𝑛 (𝑥) = 𝐴 sin(𝑛𝜋𝑥/𝐿).
- The constant A can be evaluated from the normalization condition, 𝑥=𝐿 ∫ 𝐴² sin²(𝑛𝜋𝑥/𝐿) 𝑑𝑥 = 1.
- Using the trigonometric identity sin² 𝜃 = (1 − cos² 𝜃)/2, we get 𝐴 = √(2/𝐿).
- Hence, the eigen functions are 𝛹𝑛 (𝑥) = √(2/𝐿) sin(𝑛𝜋𝑥/𝐿).
Probability Densities
- The probability of finding the particle in a small region 'dx' in a given quantum state 'n' is 2 sin²(𝑛𝜋𝑥/𝐿) 𝑑𝑥 / 𝐿.
- The probability density is 𝑃𝑛 (𝑥) = 2 sin²(𝑛𝜋𝑥/𝐿) / 𝐿.
Graphical Representation
- The eigen values 𝐸𝑛, eigen functions 𝛹𝑛 (𝑥), and probability densities 𝑃𝑛 (𝑥) can be graphically represented for n = 1, 2, and 3 quantum states.
Learn about Max Born's interpretation in quantum mechanics, where the wave function Ψ(x,t) is related to the probability density P(x,t) to determine the likelihood of finding a particle near a specific coordinate at a given time.
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