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{
"metadata": {
"name": "",
"signature": "sha256:ac0a66c98f2095bc3ebf2778dcd991b562e3ba004b2f051bfaa777d8205d2aed"
},
"nbformat": 3,
"nbformat_minor": 0,
"worksheets": [
{
"cells": [
{
"cell_type": "heading",
"level": 1,
"metadata": {},
"source": [
"Chapter 7: The Hydrogen Atom"
]
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.2, Page 248"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import scipy\n",
"import math\n",
"from scipy.integrate import quad\n",
"\n",
"#Variable declaration\n",
"a0 = 1; # For simplicity assume Bohr radius to be unity, m\n",
"\n",
"#Calculations&Results\n",
"x = lambda r:r**4*math.exp(-r/a0)\n",
"int1,err = scipy.integrate.quad(x,0,15)\n",
"y = lambda t:math.sin(t)**3\n",
"int2,err = scipy.integrate.quad(y,0,math.pi)\n",
"z = lambda p:p**0\n",
"int3,err = scipy.integrate.quad(z,0,2*math.pi)\n",
"NE = 1./(64*math.pi*a0**5)*int1*int2*int3;\n",
"print \"NE = %.f\"%NE\n",
"if round(NE) == 1:\n",
" print \"The hydrogen wave function is normalized\"\n",
"else:\n",
" print \"The hydrogen wave function is not normalized\"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"NE = 1\n",
"The hydrogen wave function is normalized\n"
]
}
],
"prompt_number": 1
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.4, Page 252"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Variable declaration\n",
"n = 3; # Principal quantum number\n",
"\n",
"#Calculations&Results\n",
"Total = 0;\n",
"print (\"\\nn l m_l 2(l + 1)\");\n",
"print (\"\\n------------------------------------\");\n",
"for l in range(0,n):\n",
" print (\"\\n%d\"% n),\n",
" print (\" %d \"% l),\n",
" if l > 0:\n",
" count = 0;\n",
" for m_l in range(-l,l+1):\n",
" print (\"%2d \"% m_l),\n",
" count = count + 1;\n",
"\n",
" if l == 1:\n",
" print(\" \"),\n",
" else:\n",
" print(\"\"),\n",
"\n",
" else :\n",
" m_l = 0;\n",
" count = 0;\n",
" print(\"%2d \"% m_l),\n",
" count = count + 1;\n",
"\n",
" print(\" %d\"% count),\n",
" Total = Total + count;\n",
"\n",
"print(\"\\n Total = %d\"% Total);\n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"n l m_l 2(l + 1)\n",
"\n",
"------------------------------------\n",
"\n",
"3 0 0 1 \n",
"3 1 -1 0 1 3 \n",
"3 2 -2 -1 0 1 2 5 \n",
" Total = 9\n"
]
}
],
"prompt_number": 5
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.7, Page 256"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"#Variable declaration\n",
"e = 1.602e-019; # Charge on an electron, C\n",
"h = 6.62e-034; # Planck's constant, Js\n",
"h_bar = h/(2*math.pi); # Reduced Planck's constant, Js\n",
"m = 9.11e-031; # Electron mass, kg\n",
"B = 2.00; # External magnetic field, T\n",
"m_l1 = 0; # Lower orbital magnetic quantum number\n",
"m_l2 = 1; # Upper orbital magnetic quantum number\n",
"\n",
"#Calculations\n",
"delta_m_l = m_l2 - m_l1; # Change in m_l\n",
"mu_B = e*h_bar/(2*m); # Bohr's magneton, J/T\n",
"delta_E = mu_B*B*delta_m_l/e; # Energy difference between components of p states of atomic hydrogen placed in the external field, eV\n",
"\n",
"#Results\n",
"print \"The value of Bohr magneton = %4.2e J/T\"%mu_B\n",
"print \"The energy difference between components of p states of atomic hydrogen placed in the external field = %4.2e eV\"%delta_E"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The value of Bohr magneton = 9.26e-24 J/T\n",
"The energy difference between components of p states of atomic hydrogen placed in the external field = 1.16e-04 eV\n"
]
}
],
"prompt_number": 6
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.8, Page 257"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import math\n",
"\n",
"#Variable declaration\n",
"m = 1.67e-027; # Mass of the proton, kg\n",
"k = 1.38e-023; # Boltzmann constant, J/K\n",
"T = 663; # Temperature of the discharge tube, K\n",
"\n",
"#Calculations\n",
"v_x = math.sqrt(3*k*T/m); # Average speed of the hydrogen atom\n",
"mu_z = 9.27e-024; # Bohr's magneton, J/T\n",
"B_grad = 1240; # Magnetic field gradient, T/m\n",
"delta_x = 0.03; # Length of the homogeneous magnetic field, m\n",
"d = 1/(2*m)*(mu_z*B_grad)*(delta_x/v_x)**2; # Separation of the atomic beam, m\n",
"\n",
"#Result\n",
"print \"The separation of the atomic beam = %4.2f mm\"%(d/1e-003)"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The separation of the atomic beam = 0.19 mm\n"
]
}
],
"prompt_number": 7
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.9, Page 259"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"#Variable declaration\n",
"n = 4; # Principal quantum number\n",
"l = 2; # For 4d-state\n",
"\n",
"#Calculations&Results\n",
"print \"\\nn l m_l m_s\"\n",
"print \"\\n------------------------------------------\"\n",
"count = 0;\n",
"for m_l in range(-l,3):\n",
" if (m_l == 0):\n",
" print \"%d\\t\"%n,\n",
" print \" %d\"%l,\n",
" print \" %d \"%m_l,\n",
" print \" +1./2, -1./2\"\n",
" else: \n",
" print \" %2d\"%m_l,\n",
" print \" +1./2, -1./2\"\n",
" count = count + 2;\n",
" \n",
"print \"Total No. of different states for 4d level of atomic hydrogen = %d\"%count\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"n l m_l m_s\n",
"\n",
"------------------------------------------\n",
" -2 +1./2, -1./2\n",
" -1 +1./2, -1./2\n",
"4\t 2 0 +1./2, -1./2\n",
" 1 +1./2, -1./2\n",
" 2 +1./2, -1./2\n",
"Total No. of different states for 4d level of atomic hydrogen = 10\n"
]
}
],
"prompt_number": 8
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.10, Page 263"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"\n",
"\n",
"def check_allowance(dn, dl, dml, dms):\n",
" if (dl == -1 or dl == 1 or dml == -1 or dml == 0 or dml == 1 or dms == -1 or dms == 0 or dms == 1) and dl != 0:\n",
" return 1;\n",
" else:\n",
" return 0;\n",
"\n",
"\n",
"state = [[2, 0, 0, 1./2],[ 3, 1, 1, 1./2],[ 2, 0, 0, 1./2],[ 3, 0, 0, 1./2],[ 4, 2, -1, -1./2],[ 2, 1, 0, 1./2]];\n",
"for i in range(0,5,2):\n",
" flag = 0; \n",
" d_n = state[i][0] - state[i+1][0];\n",
" d_l = state[i][1] - state[i+1][1];\n",
" d_m_l = state[i][2] - state[i+1][2];\n",
" d_m_s = state[i][3] - state[i+1][3];\n",
" flag = check_allowance(d_n, d_l, d_m_l, d_m_s);\n",
" if flag == 1:\n",
" print(\"\\n\\nThe transition (%d,%d,%d,1/%d) --> (%d,%d,%d,1/%d) is allowed\"%( state[i][0], state[i][1], state[i][2], state[i][3]*4, state[i+1][0], state[i+1][1], state[i+1][2], state[i+1][3]*4))\n",
" delta_E = -13.6*(1./state[i+1][0]**2-1./state[i][0]**2);\n",
" print(\"The energy of this transition is %4.2f eV\"% delta_E);\n",
" else:\n",
" print(\"\\n\\nThe transition (%d,%d,%d, %d)--> (%d,%d,%d,%d) is not allowed\"%(state[i][0], state[i][1], state[i][2], state[i][3], state[i+1][0], state[i+1][1], state[i+1][2], state[i+1][3]))\n",
"\n"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"\n",
"\n",
"The transition (2,0,0,1/2) --> (3,1,1,1/2) is allowed\n",
"The energy of this transition is 1.89 eV\n",
"\n",
"\n",
"The transition (2,0,0, 0)--> (3,0,0,0) is not allowed\n",
"\n",
"\n",
"The transition (4,2,-1,1/-2) --> (2,1,0,1/2) is allowed\n",
"The energy of this transition is -2.55 eV\n"
]
}
],
"prompt_number": 9
},
{
"cell_type": "heading",
"level": 2,
"metadata": {},
"source": [
"Example 7.13, Page 266"
]
},
{
"cell_type": "code",
"collapsed": false,
"input": [
"import scipy\n",
"from scipy.integrate import quad\n",
"import math\n",
"\n",
"#Variable declaration\n",
"a0 = 1; # For simplicity assume bohr radius to be unity\n",
"\n",
"#Calculations\n",
"p = lambda r: 4/a0**3*math.exp(-2*r/a0)*r**2\n",
"P,err = scipy.integrate.quad(p, a0, 10);\n",
"\n",
"#Result\n",
"print \"The probability of the electron in the 1s state of the hydrogen atom = %4.2f\"%P"
],
"language": "python",
"metadata": {},
"outputs": [
{
"output_type": "stream",
"stream": "stdout",
"text": [
"The probability of the electron in the 1s state of the hydrogen atom = 0.68\n"
]
}
],
"prompt_number": 10
}
],
"metadata": {}
}
]
}
|
