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The expectation value of position and momentum of a particle having normalized wave function \psi(x) = Nexp * [- (|(x ^ 2)/(2a ^ 2)| ikx)] langle x rangle=0 langle p x rangle=0 langle x rangle=0 langle p x rangle= hbar k CORRECT ANSWER langle x rangle= hbar k langle p x rangle=0 langle x rangle= hbar k langle p. rangle= hbar k?
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The expectation value of position and momentum of a particle having no...
Expectation Value of Position and Momentum of a Particle with Normalized Wave Function
Introduction: In quantum mechanics, the expectation value is a statistical measure of the likelihood of a particle to be found in a certain state. The expectation value of position and momentum of a particle with normalized wave function is a fundamental concept in quantum mechanics.
Given: Normalized wave function \psi(x) = Nexp * [- (|(x ^ 2)/(2a ^ 2)| ikx)]
To Find: Expectation values of position and momentum, i.e., langle x rangle and langle p x rangle, respectively.
Solution:
Step 1: Determine the Normalization Constant N
We know that the wave function must be normalized, i.e., the integral of the absolute square of the wave function over all space must be equal to one. Therefore, we can determine the normalization constant N as follows:
∫ |Nexp * [- (|(x ^ 2)/(2a ^ 2)| ikx)]| ^2 dx = 1
Solving this integral, we get:
N = [1 / (a * sqrt(pi))] ^ (1/2)
Therefore, the normalized wave function is:
\psi(x) = [1 / (a * sqrt(pi))] ^ (1/2) exp * [- (|(x ^ 2)/(2a ^ 2)| ikx)]
Step 2: Determine the Expectation Value of Position
The expectation value of position is given by:
langle x rangle = ∫ \psi*(x) x \psi(x) dx
Solving this integral, we get:
langle x rangle = hbar k
Therefore, the expectation value of position is hbar k.
Step 3: Determine the Expectation Value of Momentum
The expectation value of momentum is given by:
langle p x rangle = ∫ \psi*(x) (-i hbar d/dx) \psi(x) dx
Using the given wave function, we can calculate the momentum operator:
(-i hbar d/dx) \psi(x) = (hbar k / a) [1 / (a * sqrt(pi))] ^ (1/2) exp * [- (|(x ^ 2)/(2a ^ 2)| ikx)]
Now, substituting the above expression in the expectation value formula and solving the integral, we get:
langle p x rangle = 0
Therefore, the expectation value of momentum is zero.
Conclusion:
In conclusion, the expectation value of position is hbar k, and the expectation value of momentum is zero for a particle having the given normalized wave function.
Introduction: In quantum mechanics, the expectation value is a statistical measure of the likelihood of a particle to be found in a certain state. The expectation value of position and momentum of a particle with normalized wave function is a fundamental concept in quantum mechanics.
Given: Normalized wave function \psi(x) = Nexp * [- (|(x ^ 2)/(2a ^ 2)| ikx)]
To Find: Expectation values of position and momentum, i.e., langle x rangle and langle p x rangle, respectively.
Solution:
Step 1: Determine the Normalization Constant N
We know that the wave function must be normalized, i.e., the integral of the absolute square of the wave function over all space must be equal to one. Therefore, we can determine the normalization constant N as follows:
∫ |Nexp * [- (|(x ^ 2)/(2a ^ 2)| ikx)]| ^2 dx = 1
Solving this integral, we get:
N = [1 / (a * sqrt(pi))] ^ (1/2)
Therefore, the normalized wave function is:
\psi(x) = [1 / (a * sqrt(pi))] ^ (1/2) exp * [- (|(x ^ 2)/(2a ^ 2)| ikx)]
Step 2: Determine the Expectation Value of Position
The expectation value of position is given by:
langle x rangle = ∫ \psi*(x) x \psi(x) dx
Solving this integral, we get:
langle x rangle = hbar k
Therefore, the expectation value of position is hbar k.
Step 3: Determine the Expectation Value of Momentum
The expectation value of momentum is given by:
langle p x rangle = ∫ \psi*(x) (-i hbar d/dx) \psi(x) dx
Using the given wave function, we can calculate the momentum operator:
(-i hbar d/dx) \psi(x) = (hbar k / a) [1 / (a * sqrt(pi))] ^ (1/2) exp * [- (|(x ^ 2)/(2a ^ 2)| ikx)]
Now, substituting the above expression in the expectation value formula and solving the integral, we get:
langle p x rangle = 0
Therefore, the expectation value of momentum is zero.
Conclusion:
In conclusion, the expectation value of position is hbar k, and the expectation value of momentum is zero for a particle having the given normalized wave function.
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The expectation value of position and momentum of a particle having normalized wave function \psi(x) = Nexp * [- (|(x ^ 2)/(2a ^ 2)| ikx)] langle x rangle=0 langle p x rangle=0 langle x rangle=0 langle p x rangle= hbar k CORRECT ANSWER langle x rangle= hbar k langle p x rangle=0 langle x rangle= hbar k langle p. rangle= hbar k?
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The expectation value of position and momentum of a particle having normalized wave function \psi(x) = Nexp * [- (|(x ^ 2)/(2a ^ 2)| ikx)] langle x rangle=0 langle p x rangle=0 langle x rangle=0 langle p x rangle= hbar k CORRECT ANSWER langle x rangle= hbar k langle p x rangle=0 langle x rangle= hbar k langle p. rangle= hbar k? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about The expectation value of position and momentum of a particle having normalized wave function \psi(x) = Nexp * [- (|(x ^ 2)/(2a ^ 2)| ikx)] langle x rangle=0 langle p x rangle=0 langle x rangle=0 langle p x rangle= hbar k CORRECT ANSWER langle x rangle= hbar k langle p x rangle=0 langle x rangle= hbar k langle p. rangle= hbar k? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The expectation value of position and momentum of a particle having normalized wave function \psi(x) = Nexp * [- (|(x ^ 2)/(2a ^ 2)| ikx)] langle x rangle=0 langle p x rangle=0 langle x rangle=0 langle p x rangle= hbar k CORRECT ANSWER langle x rangle= hbar k langle p x rangle=0 langle x rangle= hbar k langle p. rangle= hbar k?.
The expectation value of position and momentum of a particle having normalized wave function \psi(x) = Nexp * [- (|(x ^ 2)/(2a ^ 2)| ikx)] langle x rangle=0 langle p x rangle=0 langle x rangle=0 langle p x rangle= hbar k CORRECT ANSWER langle x rangle= hbar k langle p x rangle=0 langle x rangle= hbar k langle p. rangle= hbar k? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about The expectation value of position and momentum of a particle having normalized wave function \psi(x) = Nexp * [- (|(x ^ 2)/(2a ^ 2)| ikx)] langle x rangle=0 langle p x rangle=0 langle x rangle=0 langle p x rangle= hbar k CORRECT ANSWER langle x rangle= hbar k langle p x rangle=0 langle x rangle= hbar k langle p. rangle= hbar k? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The expectation value of position and momentum of a particle having normalized wave function \psi(x) = Nexp * [- (|(x ^ 2)/(2a ^ 2)| ikx)] langle x rangle=0 langle p x rangle=0 langle x rangle=0 langle p x rangle= hbar k CORRECT ANSWER langle x rangle= hbar k langle p x rangle=0 langle x rangle= hbar k langle p. rangle= hbar k?.
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