Q2
Calculate ‹x›, ‹x2› ‹p›, ‹p2›, σx, and σp, for the nth stationary state of the infinite square well. Check that the uncertainty principle is satisfied. Which state comes closest to the uncertainty limit?
Homework Answers
The expression for wave function of an infinite square well.
Here, 
is the wave function of an infinite square well, a is the width of an infinite square well, n is the principle quantum number, x is the position of the particle.
The expression for the expectation value or average value of position
is,
Substitute 
for 
in
.
Assume that 
.
On differentiation, 
.
The limits are as follows:
When 
then 
, and when 
then 
.
The average value of position is,
Therefore, the average value of position is 
.
The expression for the expectation value or average value of 
is,
Therefore, the expectation value or average value of 
is 
.
The expectation value or average value of momentum is,
Therefore, the expectation value or average value of momentum is 
.
According to the time independent Schrodinger wave equation,
The expectation value or average value of 
is,
Substitute 
for 
in above equation as follows:
Substitute 1 for 
.
Substitute 
for 
in above equation as follows:
Therefore, the average value of 
is
.
The value of 
is,
Substitute 
for 
, and 
for 
in
.
Therefore, the value of 
is 
.
The value of 
is,
Substitute 0 for 
, and 
for 
in 
.
Therefore, the value of 
is
.
The product of 
is,
The product is smallest for the state 
.
Substitute 1 for n in
. 
Therefore, the product 
is satisfied the uncertainty principle. It comes closest to uncertainty limit at the state 
.
A) Calculate the mean values <x^2> and <p^2x> for stationery state n. B) Calculate the rms...
A) Calculate the mean values <x^2> and <p^2x> for
stationery state n.
B) Calculate the rms deviations sigma(x) and sigma(px) for
stationery state n. Check that the Heisenberg uncertainty principle
is satisfied.
1. Consider a particle of mass m in an infinite square well with potential energy 0 for 0 saSa otherwise. V (z) = For simplicity, we may take the 'universe' here to be the region of 0 sas a, which is where the wave function is nontrivial. Consequently,...
1. Consider a particle of mass m in an infinite square well with potential energy 0...
1. Consider a particle of mass m in an infinite square well with potential energy 0 for 0 Sz S a oo otherwise V (x) For simplicity, we may take the 'universe' here to be the region of 0 S z S a, which is where the wave function is nontrivial. Consequently, we may express stationary state n as where En is the associated mechanical energy. It can be shown that () a/2 and (p:)0 for stationary state n. (a)...
For an infinite square well of length L, calculate σx and σp in the n-th stationary...
For an infinite square well of length L, calculate σx and σp in the n-th stationary state. Which state has the smallest value for σxσp ?
4. Example 2.5 in your textbook points out that the operators å and p can be...
4. Example 2.5 in your textbook points out that the operators å and p can be written in terms of the ladder operators, a, (2hrnu)-12(Fip + maz) hmw 2mw Use these operators to find (x), (p), (2), (p2), and (T), for the nth stationary state of the harmonic oscillator. Check that the uncertainty principle is satisfied
Consider an infinite 1D square well with a length L with a particl of mass M...
Consider an infinite 1D square well with a length L with a particl of mass M trapped inside. (a) Calculate the expectation values(x), (2,2), (p), and (pay for the ground state where n 1. Note that the first three quantities may be calculate by using symmetry argu- ments, or by appealing to the fact that the kinetic energy is known exactly. On the other hand, to find (2) one will need to evaluate an integral. (b) Calculate ΔΧ-V(22)-(zy2 and ΔΙΕ...
1. Infinite potential quantum well. (1) Starting from the Schrödinger equation, please derive the...
1. Infinite potential quantum well. (1) Starting from the Schrödinger equation, please derive the quantized energy levels and wave functions for an infinite potential quantum well of width D 2 nm. (2) Photon emission wavelength: Please calculate the emitted photon wavelength if an electron falls from the n-2 state into n-l state inside this infinite potential quantum well. (3) Heisenberg uncertainty principle: For the n-2 state of an electron inside an infinite potential well, prove that the Heisenberg uncertainty relation...
4) A particle in an infinite square well 0 for 0
4) A particle in an infinite square well 0 for 0
The initial wave function of a free particle is: Ψ(x,0) = A, for |x| = 0,...
The initial wave function of a free particle is: Ψ(x,0) = A, for |x| = 0, otherwise where a and A are positive real numbers. The particle is in a zero (or constant) potential environment since it is a free particle a) Determine A from normalization. b) Determine φ(p) = Φ(p,0), the time-zero momentum representation of the particle state. What is Φ(p,t)? Sketch φ(p). Locate the global maximum and the zeros of φ(p). Give the expression for the zeros (i.e.,...
Parity (please answer from part a to part d) Consider Infinite Square Well Potential, V(x) =...
Parity (please answer from part a to part d)
Consider Infinite Square Well Potential,
V(x) = 0 for |x| < 1/2a and V(x) = infinity for |x| >
1/2a
a) Find energy eigenstates and eigenvalues by solving eigenvalue
equation using appropriate boundary conditions. And show
orthogonality of eigenstates.
For rest of part b to part d please look at the image below:
Problem 1 . Parity Consider an infinite square well potential, V(x) = 0 for lxl 〈 a and...
For full credit, make sure your work is clear for the grader. Show formulas, essential steps,...
For full credit, make sure your work is clear for the grader. Show formulas, essential steps, and results. 1. A particle in the infinite square well has an initial wave function given by ψ(x, 0)-(A(a-x), otherwise sa a) b) Normalize ψ(x, 0) ( that is find A) Compute (x) and (p) 2. A particle in an infinitely deep square well has a wave function given by 22tx sinir osasL 0, otherwise (a) Determine the probability of finding the particle near...
A) Calculate the mean values <x^2> and <p^2x> for
stationery state n.
B) Calculate the rms deviations sigma(x) and sigma(px) for
stationery state n. Check that the Heisenberg uncertainty principle
is satisfied.
1. Consider a particle of mass m in an infinite square well with potential energy 0 for 0 saSa otherwise. V (z) = For simplicity, we may take the 'universe' here to be the region of 0 sas a, which is where the wave function is nontrivial. Consequently,...
1. Consider a particle of mass m in an infinite square well with potential energy 0 for 0 Sz S a oo otherwise V (x) For simplicity, we may take the 'universe' here to be the region of 0 S z S a, which is where the wave function is nontrivial. Consequently, we may express stationary state n as where En is the associated mechanical energy. It can be shown that () a/2 and (p:)0 for stationary state n. (a)...
4. Example 2.5 in your textbook points out that the operators å and p can be written in terms of the ladder operators, a, (2hrnu)-12(Fip + maz) hmw 2mw Use these operators to find (x), (p), (2), (p2), and (T), for the nth stationary state of the harmonic oscillator. Check that the uncertainty principle is satisfied
Consider an infinite 1D square well with a length L with a particl of mass M trapped inside. (a) Calculate the expectation values(x), (2,2), (p), and (pay for the ground state where n 1. Note that the first three quantities may be calculate by using symmetry argu- ments, or by appealing to the fact that the kinetic energy is known exactly. On the other hand, to find (2) one will need to evaluate an integral. (b) Calculate ΔΧ-V(22)-(zy2 and ΔΙΕ...
1. Infinite potential quantum well. (1) Starting from the Schrödinger equation, please derive the quantized energy levels and wave functions for an infinite potential quantum well of width D 2 nm. (2) Photon emission wavelength: Please calculate the emitted photon wavelength if an electron falls from the n-2 state into n-l state inside this infinite potential quantum well. (3) Heisenberg uncertainty principle: For the n-2 state of an electron inside an infinite potential well, prove that the Heisenberg uncertainty relation...
4) A particle in an infinite square well 0 for 0
Parity (please answer from part a to part d)
Consider Infinite Square Well Potential,
V(x) = 0 for |x| < 1/2a and V(x) = infinity for |x| >
1/2a
a) Find energy eigenstates and eigenvalues by solving eigenvalue
equation using appropriate boundary conditions. And show
orthogonality of eigenstates.
For rest of part b to part d please look at the image below:
Problem 1 . Parity Consider an infinite square well potential, V(x) = 0 for lxl 〈 a and...
For full credit, make sure your work is clear for the grader. Show formulas, essential steps, and results. 1. A particle in the infinite square well has an initial wave function given by ψ(x, 0)-(A(a-x), otherwise sa a) b) Normalize ψ(x, 0) ( that is find A) Compute (x) and (p) 2. A particle in an infinitely deep square well has a wave function given by 22tx sinir osasL 0, otherwise (a) Determine the probability of finding the particle near...
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