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Q1 A group of researchers at the University of California at Irvine (N. Nilius, TM Wallis, and W. Ho) have constructed one dimensional Au chains of various lengths using a scanning tunnelling microscope to position individual atoms (see Fig. 1). These gold chains, ranging from a single atom to twenty atoms in length, represent one-dimensional confining potentials and their electronic properties can be understood using the simple 1D particle-in-a-box model which has been covered in the Frontiers lectures. With the STM, the researchers could measure the probability density and energy of a wide variety of electron states in the Au chains. For all questions below, assume that (i) the Au chains can be modelled as a 1D rigid box, i.e. where the potential inside the box is zero and that outside the box is infinite; (ii) the mass of the electron is 9.11 ! 10?³¹ kg (a) Write down an equation for the normalised ground state wavefunction (i.e. the n = 1 state) of a chain of Au atoms [1], and then sketch the probability density for the n=1 state of the Au? and Au?? chains, respectively†. The sketches should be on the same graph axes. [3] Fig. 1 One-dimensional chains of Au atoms constructed using a scanning tunnelling microscope (STM). [N. Nilius, TM Wallis, and W. Ho Science 297 1853 (2002)] (b) It was found that the energy difference between the n = 1 and n=2 states for a Au?? chain was 47 meV. From this result, determine the length of the Au?? chain. [5] (c) A single gold atom in a chain is 0.29 nm in diameter. For the n=1 state of the Au?? chain, what is the probability of finding an electron in the region corresponding to the gold atom at the centre of the chain? [8] (d) The STM tunnel current was measured at the centre of the Au?? chain for the n=1, 2 and 3 states respectively. For which of these measurements was the tunnel current lowest in value? Why? [3] (e) The Heisenberg uncertainty principle can be written as ?x?p ? ?. Taking a (very rough) estimate of ?x to be L/2, where L is the length of a gold chain, compare the value of the ground state energy (again for the Au?? chain) determined using the uncertainty principle with the value of E? derived from the solutions to the Schrodinger equation for the infinite potential well. [5]

          Q1 A group of researchers at the University of California at Irvine (N. Nilius, TM Wallis, and W. Ho) have constructed one dimensional Au chains of various lengths using a scanning tunnelling microscope to position individual atoms (see Fig. 1). These gold chains, ranging from a single atom to twenty atoms in length, represent one-dimensional confining potentials and their electronic properties can be understood using the simple 1D particle-in-a-box model which has been covered in the Frontiers lectures. With the STM, the researchers could measure the probability density and energy of a wide variety of electron states in the Au chains.

For all questions below, assume that (i) the Au chains can be modelled as a 1D rigid box, i.e. where the potential inside the box is zero and that outside the box is infinite; (ii) the mass of the electron is 9.11 !  10?³¹ kg

(a) Write down an equation for the normalised ground state wavefunction (i.e. the n = 1 state) of a chain of Au atoms [1], and then sketch the probability density for the n=1 state of the Au? and Au?? chains, respectively†. The sketches should be on the same graph axes. [3]

Fig. 1 One-dimensional chains of Au atoms constructed using a scanning tunnelling microscope (STM). [N. Nilius, TM Wallis, and W. Ho Science 297 1853 (2002)]

(b) It was found that the energy difference between the n = 1 and n=2 states for a Au?? chain was 47 meV. From this result, determine the length of the Au?? chain. [5]

(c) A single gold atom in a chain is 0.29 nm in diameter. For the n=1 state of the Au?? chain, what is the probability of finding an electron in the region corresponding to the gold atom at the centre of the chain? [8]

(d) The STM tunnel current was measured at the centre of the Au?? chain for the n=1, 2 and 3 states respectively. For which of these measurements was the tunnel current lowest in value? Why? [3]

(e) The Heisenberg uncertainty principle can be written as ?x?p ? ?. Taking a (very rough) estimate of ?x to be L/2, where L is the length of a gold chain, compare the value of the ground state energy (again for the Au?? chain) determined using the uncertainty principle with the value of E? derived from the solutions to the Schrodinger equation for the infinite potential well. [5]

Q1 A group of researchers at the University of California at Irvine (N. Nilius, TM Wallis, and W. Ho) have constructed one dimensional Au chains of various lengths using a scanning tunnelling microscope to position individual atoms (see Fig. 1). These gold chains, ranging from a single atom to twenty atoms in length, represent one-dimensional confining potentials and their electronic properties can be understood using the simple 1D particle-in-a-box model which has been covered in the Frontiers lectures. With the STM, the researchers could measure the probability density and energy of a wide variety of electron states in the Au chains.

For all questions below, assume that (i) the Au chains can be modelled as a 1D rigid box, i.e. where the potential inside the box is zero and that outside the box is infinite; (ii) the mass of the electron is 9.11 ! 10?³¹ kg

(a) Write down an equation for the normalised ground state wavefunction (i.e. the n = 1 state) of a chain of Au atoms [1], and then sketch the probability density for the n=1 state of the Au? and Au?? chains, respectively†. The sketches should be on the same graph axes. [3]

Fig. 1 One-dimensional chains of Au atoms constructed using a scanning tunnelling microscope (STM). [N. Nilius, TM Wallis, and W. Ho Science 297 1853 (2002)]

(b) It was found that the energy difference between the n = 1 and n=2 states for a Au?? chain was 47 meV. From this result, determine the length of the Au?? chain. [5]

(c) A single gold atom in a chain is 0.29 nm in diameter. For the n=1 state of the Au?? chain, what is the probability of finding an electron in the region corresponding to the gold atom at the centre of the chain? [8]

(d) The STM tunnel current was measured at the centre of the Au?? chain for the n=1, 2 and 3 states respectively. For which of these measurements was the tunnel current lowest in value? Why? [3]

(e) The Heisenberg uncertainty principle can be written as ?x?p ? ?. Taking a (very rough) estimate of ?x to be L/2, where L is the length of a gold chain, compare the value of the ground state energy (again for the Au?? chain) determined using the uncertainty principle with the value of E? derived from the solutions to the Schrodinger equation for the infinite potential well. [5]

Added by Julian S.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Lainey Roebuck

12/04/2024

Step 1

The wavefunction for a particle in a 1D box of length \( L \) is given by: \[ \psi_n(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{n\pi x}{L}\right) \] For the ground state (\( n=1 \)): \[ \psi_1(x) = \sqrt{\frac{2}{L}} \sin\left(\frac{\pi x}{L}\right) \] Show more…

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Q1 A group of researchers at the University of California at Irvine (N. Nilius, TM Wallis, and W. Ho) have constructed one dimensional Au chains of various lengths using a scanning tunnelling microscope to position individual atoms (see Fig. 1). These gold chains, ranging from a single atom to twenty atoms in length, represent one-dimensional confining potentials and their electronic properties can be understood using the simple 1D particle-in-a-box model which has been covered in the Frontiers lectures. With the STM, the researchers could measure the probability density and energy of a wide variety of electron states in the Au chains. For all questions below, assume that (i) the Au chains can be modelled as a 1D rigid box, i.e. where the potential inside the box is zero and that outside the box is infinite; (ii) the mass of the electron is 9.11 ! 10⁻³¹ kg (a) Write down an equation for the normalised ground state wavefunction (i.e. the n = 1 state) of a chain of Au atoms [1], and then sketch the probability density for the n=1 state of the Au₁ and Au₁₇ chains, respectively†. The sketches should be on the same graph axes. [3] (b) It was found that the energy difference between the n = 1 and n=2 states for a Au₁₇ chain was 47 meV. From this result, determine the length of the Au₁₇ chain. [5] (c) A single gold atom in a chain is 0.29 nm in diameter. For the n=1 state of the Au₁₇ chain, what is the probability of finding an electron in the region corresponding to the gold atom at the centre of the chain? [8] (d) The STM tunnel current was measured at the centre of the Au₁₇ chain for the n=1, 2 and 3 states respectively. For which of these measurements was the tunnel current lowest in value? Why? [3] (e) The Heisenberg uncertainty principle can be written as ΔxΔp ≈ ℏ. Taking a (very rough) estimate of Δx to be L/2, where L is the length of a gold chain, compare the value of the ground state energy (again for the Au₁₇ chain) determined using the uncertainty principle with the value of E₁ derived from the solutions to the Schrodinger equation for the infinite potential well. [5]