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3. Consider a Quantum mechanical Harmonic oscillator in its ground state. a) What is the expectation value of the momentum p in this state. b) In the ground state, what is the probability of finding the particle in classically allowed region? (Approximate e* 1 x x 2 the [For a quantum harmonic oscillator , (x) = mw 1/4 пh where, Ho (x) = 1, H,(x) = 2x, H2 (x) = 4x2 – 2, .] (mw "n!) 2 HnC mw 无 x) e (4 marks)?
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3. Consider a Quantum mechanical Harmonic oscillator in its ground sta...
Expectation value of momentum in the ground state:
The expectation value of an observable in quantum mechanics is given by the integral of the observable multiplied by the probability density function (wavefunction) squared. In the case of momentum, the observable is the momentum operator, which is given by p = -iħ(d/dx).
The ground state wavefunction of a harmonic oscillator is given by ψ0(x) = (mω/πħ)^(1/4) * e^(-mωx^2/2ħ).
To find the expectation value of momentum, we need to calculate the integral ∫ψ0*(pψ0) dx.
Using the momentum operator, we have pψ0(x) = -iħ(d/dx)ψ0(x).
Let's calculate it step by step:
1. Calculate ψ0*(x):
ψ0*(x) = [(mω/πħ)^(1/4) * e^(-mωx^2/2ħ)]*
2. Calculate (d/dx)ψ0(x):
(d/dx)ψ0(x) = (mω/πħ)^(1/4) * e^(-mωx^2/2ħ) * (-mωx/ħ)
3. Calculate pψ0(x):
pψ0(x) = -iħ * (mω/πħ)^(1/4) * e^(-mωx^2/2ħ) * (-mωx/ħ)
= i(mω^2/πħ^(3/2)) * x * e^(-mωx^2/2ħ)
4. Calculate the integral ∫ψ0*(pψ0) dx:
∫ψ0*(pψ0) dx = ∫[(mω/πħ)^(1/4) * e^(-mωx^2/2ħ)] * [i(mω^2/πħ^(3/2)) * x * e^(-mωx^2/2ħ)] dx
= i(mω^2/πħ^(3/2)) * ∫(mω/πħ)^(1/4) * x * e^(-mωx^2/ħ) dx
= i(mω^2/πħ^(3/2)) * ∫x * e^(-mωx^2/ħ) dx
= i(mω^2/πħ^(3/2)) * (1/2) * (ħ/mω)^(1/2)
= i(mω^2/2πħ) * (1/2) * (ħ/mω)^(1/2)
= i(mω/4π) * (1/2)
= i(mω/8π)
Therefore, the expectation value of momentum in the ground state is i(mω/8π).
Probability of finding the particle in the classically allowed region:
In the classical harmonic oscillator, the classically allowed region is where the potential energy is less than the total energy. This corresponds to the
The expectation value of an observable in quantum mechanics is given by the integral of the observable multiplied by the probability density function (wavefunction) squared. In the case of momentum, the observable is the momentum operator, which is given by p = -iħ(d/dx).
The ground state wavefunction of a harmonic oscillator is given by ψ0(x) = (mω/πħ)^(1/4) * e^(-mωx^2/2ħ).
To find the expectation value of momentum, we need to calculate the integral ∫ψ0*(pψ0) dx.
Using the momentum operator, we have pψ0(x) = -iħ(d/dx)ψ0(x).
Let's calculate it step by step:
1. Calculate ψ0*(x):
ψ0*(x) = [(mω/πħ)^(1/4) * e^(-mωx^2/2ħ)]*
2. Calculate (d/dx)ψ0(x):
(d/dx)ψ0(x) = (mω/πħ)^(1/4) * e^(-mωx^2/2ħ) * (-mωx/ħ)
3. Calculate pψ0(x):
pψ0(x) = -iħ * (mω/πħ)^(1/4) * e^(-mωx^2/2ħ) * (-mωx/ħ)
= i(mω^2/πħ^(3/2)) * x * e^(-mωx^2/2ħ)
4. Calculate the integral ∫ψ0*(pψ0) dx:
∫ψ0*(pψ0) dx = ∫[(mω/πħ)^(1/4) * e^(-mωx^2/2ħ)] * [i(mω^2/πħ^(3/2)) * x * e^(-mωx^2/2ħ)] dx
= i(mω^2/πħ^(3/2)) * ∫(mω/πħ)^(1/4) * x * e^(-mωx^2/ħ) dx
= i(mω^2/πħ^(3/2)) * ∫x * e^(-mωx^2/ħ) dx
= i(mω^2/πħ^(3/2)) * (1/2) * (ħ/mω)^(1/2)
= i(mω^2/2πħ) * (1/2) * (ħ/mω)^(1/2)
= i(mω/4π) * (1/2)
= i(mω/8π)
Therefore, the expectation value of momentum in the ground state is i(mω/8π).
Probability of finding the particle in the classically allowed region:
In the classical harmonic oscillator, the classically allowed region is where the potential energy is less than the total energy. This corresponds to the
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3. Consider a Quantum mechanical Harmonic oscillator in its ground state. a) What is the expectation value of the momentum p in this state. b) In the ground state, what is the probability of finding the particle in classically allowed region? (Approximate e* 1 x x 2 the [For a quantum harmonic oscillator , (x) = mw 1/4 пh where, Ho (x) = 1, H,(x) = 2x, H2 (x) = 4x2 – 2, .] (mw "n!) 2 HnC mw 无 x) e (4 marks)?
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3. Consider a Quantum mechanical Harmonic oscillator in its ground state. a) What is the expectation value of the momentum p in this state. b) In the ground state, what is the probability of finding the particle in classically allowed region? (Approximate e* 1 x x 2 the [For a quantum harmonic oscillator , (x) = mw 1/4 пh where, Ho (x) = 1, H,(x) = 2x, H2 (x) = 4x2 – 2, .] (mw "n!) 2 HnC mw 无 x) e (4 marks)? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about 3. Consider a Quantum mechanical Harmonic oscillator in its ground state. a) What is the expectation value of the momentum p in this state. b) In the ground state, what is the probability of finding the particle in classically allowed region? (Approximate e* 1 x x 2 the [For a quantum harmonic oscillator , (x) = mw 1/4 пh where, Ho (x) = 1, H,(x) = 2x, H2 (x) = 4x2 – 2, .] (mw "n!) 2 HnC mw 无 x) e (4 marks)? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 3. Consider a Quantum mechanical Harmonic oscillator in its ground state. a) What is the expectation value of the momentum p in this state. b) In the ground state, what is the probability of finding the particle in classically allowed region? (Approximate e* 1 x x 2 the [For a quantum harmonic oscillator , (x) = mw 1/4 пh where, Ho (x) = 1, H,(x) = 2x, H2 (x) = 4x2 – 2, .] (mw "n!) 2 HnC mw 无 x) e (4 marks)?.
3. Consider a Quantum mechanical Harmonic oscillator in its ground state. a) What is the expectation value of the momentum p in this state. b) In the ground state, what is the probability of finding the particle in classically allowed region? (Approximate e* 1 x x 2 the [For a quantum harmonic oscillator , (x) = mw 1/4 пh where, Ho (x) = 1, H,(x) = 2x, H2 (x) = 4x2 – 2, .] (mw "n!) 2 HnC mw 无 x) e (4 marks)? for GATE 2024 is part of GATE preparation. The Question and answers have been prepared according to the GATE exam syllabus. Information about 3. Consider a Quantum mechanical Harmonic oscillator in its ground state. a) What is the expectation value of the momentum p in this state. b) In the ground state, what is the probability of finding the particle in classically allowed region? (Approximate e* 1 x x 2 the [For a quantum harmonic oscillator , (x) = mw 1/4 пh where, Ho (x) = 1, H,(x) = 2x, H2 (x) = 4x2 – 2, .] (mw "n!) 2 HnC mw 无 x) e (4 marks)? covers all topics & solutions for GATE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for 3. Consider a Quantum mechanical Harmonic oscillator in its ground state. a) What is the expectation value of the momentum p in this state. b) In the ground state, what is the probability of finding the particle in classically allowed region? (Approximate e* 1 x x 2 the [For a quantum harmonic oscillator , (x) = mw 1/4 пh where, Ho (x) = 1, H,(x) = 2x, H2 (x) = 4x2 – 2, .] (mw "n!) 2 HnC mw 无 x) e (4 marks)?.
Solutions for 3. Consider a Quantum mechanical Harmonic oscillator in its ground state. a) What is the expectation value of the momentum p in this state. b) In the ground state, what is the probability of finding the particle in classically allowed region? (Approximate e* 1 x x 2 the [For a quantum harmonic oscillator , (x) = mw 1/4 пh where, Ho (x) = 1, H,(x) = 2x, H2 (x) = 4x2 – 2, .] (mw "n!) 2 HnC mw 无 x) e (4 marks)? in English & in Hindi are available as part of our courses for GATE.
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